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Professor of Astronomy in Princeton University, Author of 
"The Sun" and of a Series of Astronomical, Text-Books 

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Boston, U.S.A., and London 


Cbe ^t^enaeum press 



Two Copies Received 

JUN. 3 1902 



Entered at Stationers' Hall 

Copyright, 1902 


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The present volume has been prepared in response to a 
rather pressing demand for a text-book intermediate between 
the author's Elemeiits of Astronomy and his Greyieral Astroyiomy. 
The latter is found by many teachers to be too large for 
convenient use in the time at their disposal, while the former 
is not quite sufficiently extended for their purpose. 

The material of the new book has naturally been derived 
largely from its predecessors ; but everything has been care- 
fully worked over, rearranged and rewritten where necessary, 
and changed and added to in order to bring it thoroughly 
up to date. 

The writer is under great obligations to many persons who 
have assisted him in various ways, especially to Miss Anne S. 
Young, the head of the astronomical department in Mt. 
Holyoke College, who has carefully read and corrected all the 
proof. He is greatly indebted also to D. Appleton & Co. for 
permission to use illustrations from The Sun, to Warner & 
Swasey for photographs of astronomical instruments, and to 
numerous other friends who have kindly furnished material for 
engravings. Among these may be mentioned specially Pro- 
fessor Pickering of the Harvard Observatory, the lamented 
Keeler, and Professors Campbell and Hussey of the Lick 



Observatory, and Professors Hale, Frost, and Barnard of the 

Yerkes, besides several others to whom acknowledgment is 

made in the text. 

The volume speaks for itself as to the skillful care of 

printers and publishers in securing the most perfect mechanical 


. C. A. YOUIS^G. 

Princeton, N.J., 

April, 1902. 




CHAPTER I. — Preliminary Considerations and Defini- 
tions — Fundamental Notions and Definitions — Astronom- 
ical Coordinates and the " Doctrine of the Sphere " — The 
Celestial Globe — Exercises 6-31 

CIIAPTP]R IT. — Astronomical Instruments — Telescopes, 
and their Accessories and Mountings — Timekeepers and 
Chronographs — The Transit-Instrument — The Prime Ver- 
tical Instrument — The Almucantar — The Meridian-Circle 
and Universal Instrument — The Micrometer — The Heliom- 
eter — The Sextant — Exercises 32-65 

CHAPTER III. — Corrections to Astronomical Obser- 
vations — Dip of the Horizon — Parallax — Semidiameter 
— -■ Refraction — Twinkling or Scintillation — Twilight — 
Exercises 66-75 

CHAPTER IV. — Fundamental Problems of Practical 
Astronomy — Latitude — Time — Longitude — Azimuth 
— The Right Ascension and Declination of a Heavenly- 
Body — Exercises 76-104: 

CHAPTER V. — The Earth as an Astronomical Body — 
Its Form, Rotation, and Dimensions — Mass, Weight, and 
Gravitation — -The Earth's Mass and Density — Exercises . 105-135 

CHAPTER VL — The Orbital Motion of the Earth — 
The Apparent Motion of the Sun and the Ecliptic — The 
Orbital Motion of the Earth — Precession and Nutation — 
Aberration — The Equation of Time — The Seasons and the 
Calendar — Exercises 136-165 

CHAPTER VTI. — The Moon — The Moon's Orbital Motion 
and the Month — Distance, Dimensions, Mass, Density, and 
Force of Gravity — Rotation and Librations — Phases — 
Light and Heat — Physical Condition — Telescopic Aspect 
and Peculiarities of the Lunar Surface 166-194 




CHAPTER VIII. — The Sun — Its Distance, Dimensions, 
Mass, and Density — Its Eotation and Equatorial Accel- 
eration — Methods of studying its Surface — The Photo- 
sphere — Sun-Spots — Their ]^ature. Dimensions, Develop- 
ment, and Motions — Their Distribution and Periodicity — 
Sun-Spot Theories 195-216 

CHAPTER IX. —The Sun (Contimied) — The Spectroscope, 
the Solar Spectrum, and the Chemical Constitution of 
the Sun — The Doppler-Fizeau Principle — The Chromo- 
sphere and Prominences — The Corona — The Sun's 
Light — Measurement of the Intensity of the Sun's Heat — 
Theory of its Maintenance — The Age and Duration of 
the Sun — Summary as to the Constitution of the Sun — 
Exercises 217-260 

CHAPTER X. — Eclipses — Form and Dimensions of Shad- 
ows — Eclipses of the Moon — Solar Eclipses — Total, 
Annular, and Partial — Ecliptic Limits and Number of 
Eclipses in a Year — Recurrence of Eclipses and the Saros 

— Occultations 261-274 

CHAPTER XL — Celestial Mechanics —The Laws of 

Central Force — Circular Motion — Kepler's Laws, and New- 
ton's Verification of the Theory of Gravitation — The Conic 
Sections — The Problem of Two Bodies ^ — -The Parabolic 
Velocity — Exercises — The Problem of Three Bodies and 

Perturbations — The Tides 275-310 

CHAPTER XIL — The Planets in General — Bode's Law 

— The Apparent Motions of the Planets — The Elements of 
their Orbits — Determination of Periods and Distances — 
Perturbations, Stability of the System — Data referring to 
the Planets themselves — Determination of Diameter, Mass, 
Rotation, Surface Peculiarities, Atmosphere, etc. — Herschel's 
Illustration of the Scale of the System — Exercises . . . 311-315 

CHAPTER XIIL — The Terrestrial and Minor Planets 

— Mercury, Venus, and ]\Iars — The Asteroids — Intra- 
mercurial Planets — Zodiacal Light 346-381 

CHAPTER XIV. — The Major Planets— Jupiter : its Sat- 
ellite System; the Equation of Light, and the Distance of 
the Sun — Saturn : its Rings and Satellites — L^ranus : its 
Discovery, Peculiarities, and Satellites — Neptune : its Dis- 
covery, Peculiarities, and Satellite — Exercises 382-408 



CHAPTER XV. — Methods of detp:kmining the rAUAi.- 
LAx AND Distance of the Sun — Importance and Diffi- 
culty of the Problem — Historical — Classification of Methods 

— Geometrical Methods — Oppositions of Mars and certain 
Asteroids, and Transits of Venus — Gravitational Methods . 409-421 

CHAPTER XVI. — Comets — Their Number, Designation, 
and Orbits — Their Constituent Parts and Appearance — 
Their Spectra — Physical Constitution and Behavior — 
Danger from Comets — Exercises 422-454 

CHAPTER XVII. — Meteors and Shooting-Stars — Aero- 
lites : their Fall and Physical Characteristics ; Cause of 
Light and Heat ; Probable Origin — Shooting-Stars : their 
Number, Velocity, and Length of Path — Meteoric Shov\^ers : 
the Radiant ; Connection between Comets and Meteors — 
Exercises 45.5-476 

CHAPTER XVIII. — The Stars — Their Nature, Number, and 
Designation — Star-Catalogues and Charts — The Photo- 
graphic Campaigns — Proper Motions, Radial Motions, and 
the Motion of the Sun in Space — Stellar Parallax — Exercises 477-505 

CHAPTER XIX. — The Light of the Stars — Magnitudes 
and Brightness — Color and Heat — Spectra — Variable 
Stars — Exercises 506-536 

CHAPTER XX Stellar Systems, Clusters, and Nebul.e 

— Double and Multiple Stars — Binaries — Spectroscopic 
Binaries — Clusters — Nebulae — The Stellar Universe — 
Cosmogony — Exercises 537-673 

APPENDIX. — Transformation of Astronomical Coordinates — 
Projection and Calculation of a Lunar Eclipse — Greek 
Alphabet and Miscellaneous Symbols — Dimensions of the 
Terrestrial Spheroid — Time Constants and other Astro- 
nomical Constants 574-582 

Table I. Principal Elements of the Solar System . . 583 

Table II. The Satellites of the Solar System .... 584-585 

Table III. Comets of which Returns have been observed 586 

Table IV. Stellar Parallaxes, Distances, and Motions . 587 

Table V. Radial Velocities of Stars 588 

Table VI. Variable Stars 589 

Table VII. Orbits of Binary Stars 590 

Table VIII. :Mean Refraction 591 

INDEX 592-611 



1. Astronomy is the science which treats of the heavenly 
bodies, as is indicated by the derivation of its name (aarpov vofjio^). 
It considers : 

(1) Their motions, both real and apparent, and the laws which 
govern those motions. 

(2) Their forms, dimensions, and masses. 

(3) Their nature, constitution, and physical condition. 

(4) The effects which they produce upon each other by their 
attractions, radiations, or any other ascertainable influence. 

The earth is an immense ball, about 8000 miles in diameter, 
composed of rock and water, and covered with a thin envelope 
of air and cloud. Whirling on its axis, it rushes through 
empty space with a speed fifty times as great as that of the 
swiftest shot. On its surface we are wholly unconscious of 
the motion, because of its perfect steadiness. 

As we look up at night we see in all directions the countless The off-look 
stars ; and conspicuous among- them, and looking like stars, ^^"°^^ ^^^® 

. f.p . . earth, 

though very different in their real nature, are scattered a few 
planets. Here and there appear faintly shining clouds of light, 
like the so-called Milky Way and the nebulae, and perhaps now 
and then a comet. Most striking of all, if she happens to be 
in the heavens at the time, though really the most insignificant 
of all, is the moon. By day the sun alone is visible, flooding 
the air with its light and hiding the other bodies from the 
unaided eye, but not all of them from the telescope. 




tion of the 

Branches of 

2. The Heavenly Bodies. — The bodies thus seen from the 
earth are the heavenly bodies. For the most part they are globes 
like the earth, whirling on their axes, and moving swiftly, 
though at such distances from us that their motions can be 
detected only by careful observation. 

They may be classified as follows : First, the solar system 
proper., composed of the sun, the planets which revolve around 
it, and the satellites which attend the planets in their motion. 
The moon thus accompanies the earth. The distances between 
these bodies are enormous as compared with the size of the 
eartl ; and the sun, which rules them all, is a body of almost 
inconceivable magnitude. 

Secondly, we have the comets and the meteors., which, while 
they acknowledge the sun's dominion, move in orbits of a dif- 
ferent shape and are bodies of a very different character. 

Thirdly, we have the stars, at distances from us immensely 
greater than even those which separate the planets. The 
visible stars are suns, bodies like our own sun in nature, and like 
it, self-luminous, while the planets and their satellites shine only 
by reflected sunlight. The telescope reveals millions of stars 
invisible to the naked eye, and there are others, possibly thou- 
sands of them, that are dark and do not shine, but manifest 
their existence by effects upon their neighbors. 

Finally, there are the nebulce, of which we know very little 
except that they are cloudlike masses of shining matter, and 
belong to the region of the stars. 

3. Branches of Astronomy. — Astronomy is divided into 
many branches, some of which generally recognized are the 
following : 

(1) Descriptive Astronomy. This, as its name implies, is 
merely an orderly statement of astronomical facts and principle^ 

(2) Spherical Astronomy. This, discarding all considerations 
of absolute dimensions and distances, treats the heavenly-bodies 
simply as objects on the surface of the celestial sphere ; it deals 


only with angles and directions, and, strictly regarded, is 
merely spherical trigonometry applied to astronomy. 

(3) Practical Astronomy. This treats of the instruments, 
the methods of observation, and the processes of calculation by 
which astronomical facts are ascertained. It is quite as much 
an art as a science. 

(4) Theoretical Astronomy. This deals with the calculation 
of orbits and ephemerides, including the effect of perturbations. 

(5) Astronomical Mechanics. This is simply the application 
of mechanical principles to explain astronomical facts, chiefly the 
planetary and lunar motions. It is sometimes called " gravita- 
tional astronomy," because, with few exceptions, gravitation 
is the only force sensibly concerned in the motions of the 
heavenly bodies. 

Until about 1860 this branch of the science was generally Abandon- 
designated " physical astronomy," but the term is now objection- J^^"^ ^ 
able because of late it has been used by some writers to denote a "physical 
very different and comparatively new branch of the science, viz. : 

(6) Astronomical Physics^ or Astro- Physics. This treats of 
the physical characteristics of the heavenly bodies, their bright- 
ness and spectroscopic peculiarities, their temperature and radi- 
ation, the nature and condition of their atmospheres and surfaces, 
and all phenomena which indicate or depend on their physical 
condition. It is sometimes called The Neiv Astrononiy . 

The above branches are not distinct and separate, but overlap in all 
directions. Valuable works exist, however, bearing the different titles 
indicated above, and it is important for the student to know what subjects 
he may expect to find discussed in each, although they do not distribute 
the science between them in any strictly logical and mutually exclusive 

4. Rank of Astronomy among the Sciences. — Astronomy is 
the oldest of the natural sciences. Obviously, in the very 
infancy of the race the rising and setting of the sun, the 
phases of the moon, and the progress of the seasons must have 




still pro- 

Use in navi- 
gation and 

Use in regu- 
lation of 

Chief value 
purely intel- 

compelled the attention of even the most unobservant. Nearly 
the earliest of all existing records relate to astronomical subjects, 
such as eclipses and the positions of the planets. 

As astronomy is the oldest of the sciences, so also it is one 
of the most perfect and complete, though not in the sense that 
it has reached a maturity which admits no further development, 
for in fact it was never more vigorously alive or growing faster 
than at present. In certain aspects astronomy is also the 
noblest of the sisterhood, being the most " unselfish " of them 
all, cultivated not so much for material profit as for pure love 
of learning. 

5. Utility. — But although bearing less directly upon the 
material interests of life than the more modern sciences of 
physics and chemistry, it is really of high utility. 

It is by means of astronomy that the latitude and longitude 
of points upon the earth's surface are determined, and by such 
determinations alone is it possible to conduct extensive naviga- 
tion. Moreover, all the operations of surveying upon a large 
scale, such as the determination of international boundaries, 
depend more or less upon astronomical observations. 

The same is true of all operations which, like the railway 
service, require an accurate knowledge and observance of the 
time ; for our fundamental timekeeper is the daily revolu- 
tion of the heavens, as determined by the astronomer's transit 

At present, however, the end and object of astronomical 
study is chiefly knowledge, pure and simple. It is not likely 
that great inventions and new arts will grow out of its prin- 
ciples, such as are continually arising from chemical, physical, 
and biological studies ; but it would be rash to say that such 
outgrowths are impossible. 

The student of astronomy must, therefore, expect his chief 
profit to be intellectual, — in the widening of the range of 
thought and conception, in the pleasure attending the discovery 


of > simple law Avorking out the most far-reaching results, in the 
delight over the beauty and order revealed by the telescope 
and spectroscope in systems otherwise invisible, in the recogni- 
tion of the essential unity of the material universe and of the 
kinship between his own mind and the Infinite Reason. 

In ancient time it was believed that human affairs of every kind, the 
welfare of nations, and the life history of individuals, were controlled, or 
at least prefigured, by the motions of the stars and planets ; so that from 
the study of the heavens it ought to be possible to predict futurity. The 
pseudo-science of astrology, based upon this belief, supplied the motives that Astrology 
led to most of the astronomical observations of the ancients. As modern ^ pseudo- 
chemistry had its origin in alchemy, so astrology was the progenitor of 
astronomy, and it is remarkable how persistent a hold,this baseless delusion 
still retains upon the credulous. 

6. Place in Education Apart from the utility of astronomy 

in the ordinary sense of the word, the study of the science is of 

high value as an intellectual training. Na other so operates Educational 

to give us, on the one hand, just views of our real insignificance ^^^^^®- 

in the universe of space, matter, and time, or to teach us, on 

the other hand, the dignity of the human intellect as being the 

offspring, and measurably the counterpart, of the Divine, — able 

in a sense to comprehend the universe and understand its plan 

and meaning. 

The study of the science cultivates nearly every faculty of 
the mind ; the memory, the reasoning power, and the imagina- 
tion all receive from it special exercise and development. By 
the precise and mathematical character of many of its discussions 
it enforces exactness of thought and expression, and corrects 
the vague indefi-uiteness which is apt to be the result of purely 
literary training ; while, on the other hand, by the beauty and ^ 
grandeur of the subjects which it presents, it stimulates the 
imagination and gratifies the poetic sense. 



Fundamental Notions and Definitions — Astronomical Coordinates and the "Doctrine 

of the Sphere " — The Celestial Globe 

Astronomy, like all the other sciences, has a terminology of 
its own, and uses technical terms in the description of its facts 
and phenomena. In a popular work it would be proper to 
avoid such terms as far as possible, even at the expense of 
circumlocutions and occasional ambiguity; but in a text-book it 
is desirable that the student should be introduced to the most 
important of them at the very outset and be made sufficiently 
familiar with them to use them intelligently and accurately. 

7. The Celestial Sphere.^ — The sky appears like a hollow 
vault, to which the stars seem to be attached, like gilded nail- 
heads upon the inner surface of a dome. We cannot judge 
of the distance to this surface from the eye further than to 
perceive that it must be very far away; it is therefore natural 
and extremely convenient to regard the distance of the sky 
The celestial as everywhere the same and unlimited. The celestial sphere^ 
sphere ^^ ^^ ^g Called, is couceived of as so enormous that the whole 


as infinite, material universe of stars and planets lies in its center like a 
few grains of sand in the middle of the dome of the Capitol. 
Its diameter, in technical language, is taken as mathematically 
infinite^ i.e.^ greater than any assignable quantity. 

Since the radius of the sphere is thus infinite, it follows that 
all the lines of any set of parallels will appear, if produced 

1 The study of the celestial sphere and its circles is greatly aided by the use 
of a globe or armillary sphere. Without some such apparatus it is rather difficult 
for a beginner to get clear ideas upon the subject. 



D C 

indefinitely, to pierce it at a single point, the vanishing point 
of perspective, or the point at infinity of projective geometry. 
However far apart the lines may be and whatever, therefore, 
may be the distances in miles between the points at which they 
pierce the surface of the celestial sphere, yet, seen by the 
observer at its infinitely distant center, the angular distance 
between those points is utterly insensible, and they coalesce 
into one. Thus the axis of the earth and all lines parallel to 
it .pierce the heavens at one point, the celestial pole ; and the 
plane of the earth's equator, keep- 
ing parallel to itself during her 
annual circuit around the sun, 
marks out only one celestial equa- 
tor in the sky. 

8. The place of a heavenly body 
is simply the point where a line 
drawn from the observer through "f^ 
the body in question and continued 
onward pierces the celestial sphere. 
It depends solely upon the direc- 
tion of the body and has nothing 
to do with its distance. ^ Thus, in Fig. 1 A^ B, C, etc., are the 
apparent places of a, h, c, etc., the observer being at 0. Objects 
that are nearly in line with each other, as A, z, k, will appear 
close together in the sky, however great the real distance between 
them. The moon, for instance, often looks to us very near a 
star, which is always at an immeasurable distance beyond her. 

9. Linear and Angular Dimensions and Measurement. - — Linear 
dimensions are such as can be expressed in linear units ; i.e., 
in miles, feet, or inches ; kilometers, meters, or millimeters. 
Angular dimensions are expressed in angular units ; ^.e., in degrees, 
minutes, and seconds, or sometimes in radians, the radian being 
the angle which is measured by an arc equal in length to the 
radius, determined by dividing the circumference by 2 tt. 

of parallels 
to a single 
l)oint on the 

Fig. 1 

Place of a 
body de- 
pends solely 
on its direc- 
tion from 

Value of the 
radian in 
and seconds. 



units used 
in express- 
ing measure- 
ments on 

The radian, therefore, equals 57°. 29 {i.e., 360° -^ 2 tt), 
or 3437'.75 {i.e., 21600' ^ 2 tt), 
or 206264'^8 {i.e., 1 296000'' ^ 2 7r). 

Hence, to reduce to seconds of arc an angle expressed in radians, 
we must multiply its value in radians hy 20626 If. 8 ; a relation of 
which we shall make frequent use. 

Obviously, angular units alone can properly be used in describ- 
ing apparent distances in the sky. One cannot say correctly 
that the two stars known as "the pointers" are so many/ee^ 
apart ; their distance is about five degrees. 

It is very important that the student of astronomy should 
accustom himself as soon as possible to estimate celestial meas- 
ures in angular units. A little practice soon makes it easy, 
although the beginner is apt to be embarrassed by the fact that 
the sky appears to the eye to be not a true hemisphere, but a 
flattened vault, so that all estimates of angular distances for 
objects near the horizon are apt to be exaggerated. The moon 
when rising or setting looks to most persons much larger than 
when overhead, and the " Dipper-bowl " when underneath the 
pole seems to cover a much larger area than when above it. 

Apparent This illusion (for it is merely an illusion), which makes the sun and 

enlargement heavenly bodies when near the horizon appear larger than when high up 
in the sky, is probably due to the fact that in the latter ca^e we have no 
intervening objects by which to estimate the distance, and it therefore is 
judged to be smaller than at the horizon. If we look at the sun or moon 
when near the horizon through a lightly smoked glass which cuts off the 
view of the landscape, the object immediately shrinks to its ordinary size. 

of sun and 
moon near 
the horizon. 

radius, and 
diameter of 
a globe. 

10. Relation between the Distance and Apparent Size of an 
Object. — Suppose a globe having a (linear) radius BC equal to 
r. As seen from the point A (Fig. 2) its apparent {i.e., angular) 
semidiameter will be BAG or s, its distance being AC or B. 

We have immediately, from trigonometry, since ^ is a right 
angle, sin s = r /B, whence also r = B X sin s, and B = r ^ sin s. 


If, as is usual in astronomy, the diameter of an object is 
small as compared with its distance, so that sin s practically 

A -Ts 

< — B — ^ 

Fig. 2 

equals s itself, we may write s = r/R, which gives s in radians 
(not in degrees or seconds). If we wish to have it in the 
ordinary angular units, 

s° = 57.3r/^; or s' = 3437.7 r/^ ; or s" = 206264.8 r/^; 

also R = 206264.8 r/s", 

where s° means s in degrees ; s', in minutes of arc ; s", in seconds 
of arc. 

In either form of the equation we see that the apparent 
diameter varies directly as the linear diameter and inversely as 
the distance. 

In the case of the moon, i^ = about 239000 miles; and r, 
1081 miles. Hence s (in radians) = 2 3 9 oi"o ~ six ^^ ^ radian, 
which is about 933", — a little more than one fourth of a 

It may be mentioned here as a rather curious fact that to most persons Apparent 

the moon, when at a considerable altitude, appears about a foot in diam- distance of 

eter ; — at least, this seems to be the average estimate. This implies that surtace 

of tliG cgIgs™ 
the surface of the sky appears to them only about 110 feet away, since ^- 1 1 

that is the distance at which a disk one foot in diameter would have an 

angular diameter of jfo of a radian, or i°. 

Probably this is connected with the physiological fact that our muscular 

sense enables us to judge moderate distances pretty fairly up to 80 or 100 

feet, through the " binocular parallax " or convergence of the eyes upon 

the object looked at. Beyond that distance the convergence is too slight 

to be perceived. It would seem that we instinctively estimate the moon's 

distance as small as our senses will permit when there are no intervening 

objects which compel our judgment to put her further off. 





In order to be able to describe intelligently the position of a 
heavenly body in the sky, it is convenient to suppose the inner 
surface of the celestial sphere to be marked off by circles traced 
upon it, — imaginary circles, of course, like the meridians and 
parallels of latitude upon the surface of the earth. 

Three distinct systems of such circles are made use of in 
astronomy, each of which has its own peculiar adaptation for its 
special purposes. 

ical zenith 
and nadir 



11. The Zenith and Nadir. — If we suspend a plumb-line, and 
imagine the line extended upward to the sky, it will pierce 
the celestial sphere at a point directly overhead, known as the 
Astronomical Zenith, or the Zenith simply, unless some other 
qualifier is annexed. 

As will be seen later (Sec. 130, 5), the plumb-line does not 
point exactly to the center of the earth, because the earth 
rotates on its axis and is not strictly spherical. If an imagi- 
nary line be drawn from the center of the earth upward 
through the observer, and produced to the celestial sphere, 
it marks a different point, known as the geocentric zenith, 
which is never very far from the astronomical zenith, but 
must not be confounded with it. 

For most purposes the astronomical zenith is the better 
practical point of reference, because its position can be deter- 
mined directly by observation, which is not the case with the 


The Nadir is the point opposite to the zenith, directly under 
foot in the invisible part of the celestial sphere. 

Both "zenith" and "nadir" are derived from the Arabic, as are many 
other astronomical terms. It is a reminiscence of the centuries when the 
Arabs were the chief cultivators of science. 

12. The Horizon. — If now we imagine a great circle drawn The horizon 
completely around the celestial sphere half-way between the ^ ^^ " 
zenith and nadir, and therefore 90° from each of them, it will 

be the Horizon (pronounced ho-ri'-zon, not hor'-i-zon). 

Since the surface of still water is always perpendicular to 
the direction of gravity,, we may also define the horizon as the 
great circle in which a plane tangent to a surface of still water 
at the place of observation cuts the celestial sphere. 

Many writers distinguish between the sensible and rational Uuueces- 
horizons, — the former beinsf defined by a horizontal plane drawn ^^^'^ distmc- 

■ *^ _ -^ tion between 

through the observer's eye, while the latter is defined by a plane sensible and 
parallel to this, but drawn through the center of the earth, ^'^tionai 
These two planes, however, though 4000 miles apart, coalesce 
upon the infinite celestial sphere into a single great circle 90° 
from both zenith and nadir, agreeing with the first definition 
given above. The distinction is unnecessary. 

13. Visible Horizon. — The word " horizon " (from the Greek) The visible 
means literally ''the boundary" — that is, the limit of the land- ^°^'^^°^- 
scape, where sky meets earth or sea; and this boundary line 

is known in astronomy as the visible horizon. On land it is 
of no astronomical importance, being irregular; but at sea it 
is practically a true circle, nearly coinciding with the horizon 
above defined, but a little below it. When the observer's eye 
.is at the water-level, the coincidence is exact ; but if he is 
at an elevation above the surface, the line of sight drawn 
from his eye tangent to the water inclines or dips down, on 
account of the curvature of the earth, by a small angle known 
as the dip of the horizon^ to be discussed further on (Sec. 77). 



and. the 

and zenith- 

14. Vertical Circles ; the Meridian and the Prime Vertical. — 

Vertical circles are great circles drawn from the zenith at right 
angles to the horizon, and therefore passing through the nadir 
also. Their number is indefinite. 

That particular vertical circle which passes north and south 
through the pole, to be defined hereafter, is known as the Celes- 
tial Meridian^ and is evidently the circle traced upon the celestial 
sphere by the plane of the terrestrial meridian upon which the 
observer is located. The vertical circle at right angles to 
the meridian is called the Prime Vertical. The points where 


Fig. 3. — The Horizon and Vertical Circles 

O, the place of the observer. 
OZ, the observer's vertical. 
Z, the zenith ; P, the pole. 
SWNE, the horizon. 
SZPN, the meridian. 
EZW, the prime vertical. 

M, some star. 

ZMH, arc of the star's vertical circle. 

TMR, the star's almncantar. 

Angle TZM, or arc SH, star's azimuth. 

Arc HM, star's altitude. 

Arc ZM, star's zenith-distance. 

the meridian intersects the horizon are the north and south 
points; and the east and west points are midway between them. 
These are known as the Cardinal Points. 

The parallels of altitude., or ahnucantars, are small circles of 
the celestial sphere drawn parallel to the horizon, sometimes 
called circles of equal altitude. 

15. Altitude and Zenith-Distance. — The Altitude of a heav- 
enly body is its angular elevation above the horizon, ^.e., 
the number of degrees between it and the horizon, measured 
on a vertical circle passing through the object. In Fig. 3 the 


vertical circle ZMH passes through the body 31. The arc MH 
is the altitude of Jf, and the arc ZM (the complement of MH) 
is its zenith-distance. 

16. Azimuth. — The Azimuth (an Arabic word) of a heavenly Azimuth 
body is the same as its " bearing " in surveying ; measured, ^^^^6^- 
however, from the true meridian and not from the magnetic. 

It may be defined as the angle formed at the zenith between 
the meridian and the vertical circle which passes through the 
object; or, what comes to the same thing, it is the arc of the 
horizon intersected between the south point and the foot of this 

In Fig. 3 SZM is the azimuth of M, as is also the arc SIT, 
which measures this angle. The distance of H from the east or 
west point of the horizon is called the amplitude of the body, Amplitude 
but the term is seldom used except in describing the point ^©^^ed. 
where the sun or moon rises or sets. 

There are various ways of reckoning azimuth. Formerly Metiiod of 
it was usually expressed in the same way as the " bearingf " in I'eckonmg 

. (. azimuth. 

surveying; i.e., so many degrees east or west of north or 
south. In the figure, the azimuth of M thus expressed is 
about S. 50° E. The more usual way at present, however, 
is to reckon it from the south point clear around through the 
west to the point of beginning, so that the arc SWNKEH 
would be the azimuth of Jf, — about 310°. 

17. Altitude and azimuth are for many purposes incon- inconven- 
venient, because they continually change for a celestial obiect. ^®^^® °^ ^^^^' 

■^ , . . tude and 

It is desirable, therefore, in defining the place of a body in the azimuth, 
heavens, to use a different way which shall be free from this 
objection ; and this can be done by taking as the fundamental 
line of our system, not the direction of gravity, which is differ- 
ent at any two different points on the earth's surface and is 
continually changing as the earth revolves, but the direction 
of the eartKs axis., which is practically constant. 




Apparent 18. The Apparent Diurnal Rotation of the Heavens. — If on 

the heavens. ^'^^^ clear evening in the early autumn, say about eight o'clock 
on the 2 2d of September, we face the north, we shall find the 

Fig. 4. — The Northern CircumiDolar Constellations 

appearance of that part of the heavens directly before us sub- 
stantially as shown in Fig. 4. In the north is the constellation 
of the Great Bear (Ursa Major), characterized by the conspicu- 
ous group of seven stars, known as the Great Dipper, which 
lies with its handle sloping upward to the west. The two 


easternmost stars of the four which form its bowl are called the 
pointers^ because they point to the pole-star^ — a solitary star The poie- 
not quite half-way from the horizon to the zenith (in the latitude ^*^^ ^^^ *^® 

^ 'J ^ ^ pointers. 

of New York), and about as bright as the brighter of the two 
pointers. It is often called Polaris. 

High up on the opposite side of the pole-star from the Great 
Dipper, and at nearly the same distance, is an irregular zigzag 
of five stars, each about as bright as the pole-star itself. This 
is the constellation of 

If now we watch these 
stars for only a few 
hours, we shall find that 
while all the configura- 
tions remain unaltered, 
their places in the sky 
are slowly changing. 
The Dipper slides down- 
ward towards the north, 
so that by eleven o'clock 
the pointers are directly 
under Polaris. Cassio- 
peia still keeps Oppo- Fig. 5. - Polar star Trails 

site, however, rising towards the zenith ; and if we were to 
continue to watch them the whole night, we should find that 
all the stars appear to be moving in circles around a point near 
the pole-star, revolving in the opposite direction to the hands 
of a watch (as we look up towards the north), with a steady 
motion which takes them completely around once a day, or, 
to be exact, once in the sidereal day^ consisting of 23^56"'4^1 
of ordinary time. They behave just as if they were attached 
to the inner surface of a huge revolving sphere. 

Instead of watching the stars with the eye, the same result 
can be still better reached by photography. A camera is pointed 



Polar star 


of the poles 
of rotation. 

of the pole. 

up towards the pole-star and remains firmly fixed while the stars, 
by their diurnal motion, impress their " trails " upon the plate. 
Fig. 5 is copied from a negative made by the author with an 
exposure of about three hours. 

If instead of looking towards the north we now look south- 
ward, we shall find that there also the stars appear to move in 
the same kind of way. All that are not too near the pole-star 
rise somewhere in the eastern horizon, ascend not vertically but 
obliquely to the meridian, and descend obliquely to their setting 
at points on the western horizon. The motion is always in an 
arc of the circle, called the star's diurnal circle^ the size of w^hich 
depends upon the star's distance from the pole. Moreover, all 
these arcs are strictly parallel. 

The ancients accounted for these obvious facts by supposing 
the stars actually fixed upon a real material sphere, really 
turning daily in the manner indicated. According to this view 
there must therefore be upon the sphere two opposite, pivotal 
points which remain at rest, and these are the j^oles. 

19. Definition of the Poles. — The Celestial JPoles, or Poles of 
notation (when it is necessary, as sometimes happens, to dis- 
tinguish between these poles and the poles of the ecliptic), may 
therefore be defined as those two points in the shy where a star 
would have no diurnal motion. The exact position of either 
pole may be determined with proper instruments by finding the 
center of the small diurnal circle described by some star near 
it, as for instance by the pole-star. 

Since the two poles are diametrically opposite in the sky, only one of 
them is usually visible from a given place ; observers north of the equator 
see only the north pole, and vice versa in the southern hemisphere. 

Knowing as we now do that the apparent revolution of the 
celestial sphere is due to the real rotation of the earth on 
its axis, we may also define the poles as the two points where 
the eartKs axis of rotatiori (or any set of lines parallel to it), 
produced indefinitely, would pierce the celestial sphere. 


20. The Celestial Equator, or Equinoctial, and Hour-Circles. — 

The Celestial Equator is the great circle of the celestial sphere^ 
drawn half-way between the poles (therefore everywhere 90° from 
each of them), and is the great circle in which the plane of the 
earth's equator cuts the celestial sphere (Fig. 6). It is often 
called the Equinoctial. Small circles drawn parallel to the 
equinoctial, like the parallels of latitude on the earth, are called 
parallels of declination^ a star's 
parallel of declination being iden- 
tical with its diurnal circle. 

The great circles of the celestial 
sphere which pass through the poles, 
like the meridians on the earth, 
and are therefore perpendicular 
to the celestial equator, are called 
Hour- Circles. On celestial globes 
twenty-four of them are usually 
drawn, corresponding one to each 
of the twenty-four hours, but the 
real number is indefinite ; an 
hour-circle can be drawn through 
any star. That particular hour-circle tvhich at any moment 
passes through the zenith of the observer^ coincides with the 
celestial meridian, already defined. 

21. Declination and Hour Angle. — The Declination of a star 
is its distance in degrees north or south of the celestial equator ;' 
-\- if north, — if south. It corresponds precisely with the lati- 
tude of a place on the earth's surface, but cannot be called 
celestial latitude, because the term has been preoccupied by an 
entirely different quantity to be defined later (Sec. 27). 

The Hour Angle of a star at any moment is the angle at 
the pole between the celestial meridian and the hour-circle of the 
star. In Fig. 7, for the body m it is the angle mPZ, or the 
arc Q Y. 

of the 

Parallels of 
with diurnal 

Fig. 6.— The Plane of the Earth's 
Equator produced to cut the Celes- 
tial Sphere 

defined. The 
as an 


Hour angle 



Relation of 
units of 
time to units 
of angle. 

This angle, or arc, may of course be measured like any other, 
in degrees, but since it depends upon the time which has elapsed 
since the body was last on the meridian, it is more usual to 
measure it in hours, minutes, and seconds of time. The hour 
is then equivalent to ^^ of a circumference, or 15°, and the 
minute and second of time to 15' and 15'' of arc, respectively. 
Thus, an hour angle of 4^2""3« equals 60° 30' 45". 


Fig. 7. — Hour-Circles, etc. 

O, place of the observer ; Z, his zenith. 

SENW, the horizon. 

POP', the axis of the celestial sphere. 

P and P', the two poles of the heavens. 

EQWT, the celestial equator, or equinoc- 

X, the vernal equinox, or " first of Aries." 

PXP', the equinoctial colure, or zero hour- 

m, some star. 

Ym, the star's declination ; Pin, its north- 
polar distance. 

Angle mPi?= arc QY, the star's (eastern) 
hour angle ; = 24h minus star's western 
hour angle. 

Angle XPm = arc X F, star's right ascension. 
Sidereal time at the moment = 24'^ minus 
angle XPQ. 

The position of the body m (Fig. 7) is, then, perfectly defined 
by saying that its declination is +25° and its hour angle 40° east 
(or simply 320°, if we choose, as is usual, to reckon completely 
around in the direction of the diurnal motion). Instead of 40 
degrees, we might say 2M0"^ of time east, or simply 21^20™ to 
correspond to the 320°. 

22. The declination of a star, omitting certain minutise for 
the present, remains practically unaltered even for years, but 


the hour angle changes continually and uniformly at the rate of The hour 
15° for every sidereal hour. This unfits it for use in ephemeri- ^^^^^ 

. changes 

des or star-catalogues. We must substitute for the meridian continually 
some other hour-circle passing through a well-defined point with the 
which participates in the diurnal rotation and so retains an 
unchanging position relative to the stars. Such a point, 
selected by astronomers nearly two thousand years ago, is the 
so-called Vernal Equinox^ or First of Aries. 

23. The Ecliptic, Equinoxes, Solstices, and Colures. — The 
sun, moon, and planets, though apparently carried by the 
diurnal revolution of the celestial sphere, are not, like the stars, 
apparently fixed upon it, but move over its surface like glow- 
worms creeping on a whirling globe. In the course of a year, 
as will be explained later (Sec. 156), the sun makes a complete 
circuit of the heavens, traveling among the stars in a great 
circle called the Ecliptic. The ecliptic. 

The ecliptic, cuts the celestial equator in two opposite points 
at an angle of about 23i°. These points are the equinoxes. 
The Vernal Equinox^ or First of Aries (symbol ©f ), is the point The vernal 
where the sun crosses from the south to the north side of the ®^"^^o^- 
equator^ on or about the 21st of March. The other is the 
autumnal equinox. 

The summer and winter Solstices are points on the ecliptic, The soi- 
midway between the two equinoxes and 90° from each, where ^*^^®^' 
the sun attains its maximum declination of H- 23i° and — 23i° 
in summer and winter, respectively. 

The hour-circles drawn from the pole (of rotation) through 
the equinoxes and solstices are called the equinoctial and 

solstitial Colures. ^ The colures. 

Neglecting for the present the gradual effect of pre- 
cession (Sec. 165), these points and circles are fixed with 
reference to the stars, and form a framework by which the 
places of celestial objects may be conveniently defined and 



Position of 
the vernal 

of right 

The sidereal 


of sidereal 

No conspicuous star marks the position of the vernal equinox; 
but a line drawn from the pole-star through /? Cassiopeise and 
continued 90° from the pole will strike very near it. 

24. Right Ascension.— The Right Ascension of a star may 
now be defined as the angle made at the celestial pole between 
the hour-circle of the star and the hour-circle ivhich passes 
through the venial equinox (called the equinoctial colure), or 
as the arc of the celestial equator intercepted between the vernal 
equinox and the point where the starts hour-circle cuts the equator. 
Right ascension is reckoned always eastward from the equinox, 
completely around the circle, and may be expressed either in 
degrees or in time units. A star one degree west of the equinox 
has a right ascension of 359°, or 23^56™. 

Evidently the diurnal motion does not affect the right ascen- 
sion of a star, but, like the declination, it remains practically 
unchanged for years. In Fig. 7 (Sec. 21), if X be the vernal 
equinox, the right ascension of m is the angle ^Fm, or the arc 
XZ measured from ^X eastward. 

25. Sidereal Day and Sidereal Time. — The sidereal da^ is 
the interval of time between two successive passages of a fixed 
star over a given meridian, and at any place it begins at the 
moment when the vernal equinox is on the meridian; it is 
about four minutes shorter than the solar day, and like it is 
divided into twenty-four (sidereal) hours with corresponding 
sidereal minutes and seconds, all shorter than the corresponding 
solar units. 

The sidereal time at any moment is the time shown by a 
clock so set and regulated as to show zero hours, zero minutes, 
and zero seconds at the moment when the vernal equinox 
crosses the meridian. It is the hour angle of the vernal equinox, 
or, what is the same thing, the right ascension of the observer's 

26. Observatory Definition of Right Ascension. — The right 
ascension of a star may now be correctly, and for observatory 


purposes, most conveniently defined as the sidereal time at the Observatory 
moment when the star is crossing the observer's meridian. Since ^^^^^^^^^^ ^^ 

•^ ^ _ right ascen- 

tlie sidereal clock indicates zero hours at sidereal noon, i.e., at sion. 
the moment when the vernal equinox is on the meridian, its 
face at any other time shows the hour angle of the equinox ; 
and this is what has just been defined as the right ascension of 
all stars which may then happen to be on the meridian (common 
to them all since they all lie on the same hour-circle). 



27. Celestial Latitude and Longitude — The ancient astrono- Definition 
mers confined their observations mostly to the sun, moon, and ®^ celestial 

1 T • 1 c latitude and 

planets, which are never lar irom the ecliptic, and for this longitude, 
reason the ecliptic (which is simply the trace of the plane of the 
earth's orbit upon the celestial sphere) was for them a more 
convenient circle of reference than the equator, — especially as 
they had no accurate clocks. According to their terminology, 
Latitude (celestial) is the angular distance of a heavenly body 
north or south of the ecliptic; Longitude (celestial) is the arc 
of the ecliptic intercepted between the vernal equinox (^) and 
the foot of a circle draum from, the pole of the ecliptic to the 
ecliptic through the object. Longitude, like right ascension, is 
always reckoned eastward from the equinox. 

Circles drawn from the poles of the ecliptic perpendicular to 
the ecliptic are called secondaries to the ecliptic, — by some Secondaries 
writers " ecliptic meridians," and on some celestial gflobes are ^^ ^^^ 

^ _ ^ ecliptic. 

drawn instead of hour-circles. 

The poles of the ecliptic are the points 90° distant from the Poles of the 
ecliptic. The position of the north ecliptic pole is shown in ^^iiptic. 
Fig. 4. It is on the solstitial colure, about 231-° distant from 
the pole of rotation, in declination QQ}y° and right ascension 
18*\ It is marked by no conspicuous star. 



It is unfortunate, or at least confusing to beginners, that celes- 
tial latitude and longitude should not correspond with the ter- 
restrial quantities that bear the same name. Great care must 
be taken to observe the distinction. 

The gravity- 
system of 

The two 
which de- 
pend upon 
the rotation 
of the earth. 

28. Recapitulation. — The direction of gravity at the point 
where the observer happens to stand determines the zenith and 
nadir^ the horizon and the almucantars, or parallels of altitude, 
and all the vertical circles. One of the verticals, the meridian, 
is singled out from the rest by the circumstance that it is the 
projection of the observer'' s terrestrial meridian upon the celestial 
sphere and passes through the pole, marking the north and south 
points where it cuts the horizon. Altitude and azimuth, or their 
complements, zenith-distance and amplitude, define the position 
of a body by reference to the horizon and meridian. 

This set of points and circles shifts its position among the 
stars with every change in the place of the observer and every 
moment of time. Each place and hour has its own zenith, its 
own horizon, and its own meridian. 

In a similar way, the direction of the eartKs axis, which is 
independent of the observer's place on the earth, determines the 
pole (of rotation), the equator, parallels of declination, and the 
hour-circles. Two of these hour-circles are singled out as 
reference lines: one of them is the hour-circle which at any 
moment passes through the zenith and coincides with the merid- 
ian, — a purely local reference line ; the other, the equinoctial 
colure, which passes through the vernal equinox, a point chosen 
from its relation to the sun's annual motion. 

Declination and hour angle define the place of a star with 
reference to the equator and meridian, while declination and 
right ascension refer it to the equator and vernal equinox. The 
latter are the coordinates usually given in star-catalogues and 
almanacs for the purpose of defining the position of stars and 
planets, and they correspond exactly to latitude and longitude on 


the eartli^ by means of which geographical positions are desig- 
nated. Of in the sky takes the place of Greenwich on the earth. 

Finally, the earth's orbit gives us the great circle of the sky The ecliptic 
known as the ecliptic ; and celestial latitude and longitude define ^y^tem. 
the position of a star with reference to the ecliptic and the ver- 
nal equinox (of ). For most purposes this pair of coordinates is 
practically less convenient than right ascension and declination ; 
but it came into use centuries earlier, and has advantages in 
dealing with the planets and the moon. 

29. The scheme given below presents in tabular form the 
relations of the four different systems to each other. In each 
case one of the two coordinates is measured along a primary 
circle, from a point selected as the origin^ to a point where a 
secondary circle cuts it, drawn through the object perpendicular 
to the primary. The second coordinate is the angular distance 
of the object from the primary circle measured along this 

exhibit of 
the four 
systems of 


Primary Cir- 
cle, HOW 





^ SI 


Direction of 


South point 
on horizon 

Vertical cir- 
cle of star 



' 1 

Rotation of 


Foot of the 

meridian on 


of star 

Hour angle 



Rotation of 


The vernal 
equinox (°f ) 

of star 

Right ascen- 


Plane of 
earth's orbit 


The vernal 
equinox (°f ) 

Secondary to 

through star 


30. Relation of the Coordinates on the Sphere. — Fig. 8 shows how these Diagram 

coordinates are related to each other. The reader is supposed to be showing the 

looking down on the celestial sphere from above, the circle SENWA being ^'elation of 
the horizon. 

the systems. 



. Z is the zenith ; P, the north pole (of rotation) ; P', the pole 
of the ecliptic ; °f , the vernal equinox, and dZh, the autumnal ; 8^ E, N, W 
are the cardinal points of the horizon. The oval W°fMQCE:QzR is 
the celestial equator, and the narrower one, ^LB:C^K, is the ecliptic. 
The angle B<^C, measured by the arcs BC and PP', is the obliquity of the 
ecliptic, for which the usual symbol is e or e. 

is some celestial object. Then the arc A (projected as a straight 
line) is its altitude and the angle OZS its azimuth. OM is its declination 

Fig. 8. —Relation of the Different Coordinates 

and OPQ its hour angle. J'PM is its right ascension = arc f M. OL is its 
latitude s^nd f P'L (= arc T L) is its longitude. T ^ is 90° of the equi- 
noctial colure and P'PBC is 90° of the solstitial colure. The angles T P'B 
and fPC are each 90°. 

For methods and formulae by which either set of coordinates may be 
"transformed" into one of the others, see Sees. 700 and 701 (Appendix). 


31. The Astronomical Triangle. — The triangle PZO (pole- 
zenith-object) (Fig. 8) is often called the astronomical triangle 
because so many problems, especially of nautical astronomy, 
depend on its solution. Its sides and angles are all named, — 
PZ is the colatitude of the observer, ZO is the zenith-distance of 
the object, and OF is its north polar distance, or complement of 
its declination. The angle F is the hour angle of the object, the 
angle Z is the supplement of its azimuth, and, finally, the angle 
at is called the parallac- 
tic angle, because it enters 
into the calculations of the 
effects of parallax and re- 
fraction upon the right 
ascension and declination 
of a body. Any three of 
the parts being given the 
others can, of course, be 

32. Relation of the Place 
of the Celestial Pole to the 
Observer's Latitude. — If 
an observer were at the north pole of the earth, it is clear that 
the pole-star would be very near his zenith, while it would be 
at his horizon if he were at the equator. The place of the pole 
in the sky, therefore, depends entirely on the observer's latitude, 
and in this very simple way the altitude of the pole (its height 
in degrees above the horizon) is always equal to the latitude 
of the observer. This will be clear from Fig. 9. The latitude 
(astronomical) of a place may be defined as the angle between the 
direction of gravity at that place and the plane of the earth's 
equator, — the angle ONQ in Fig. 9. If at we draw -fTi^T' per- 
pendicular to ON, it will be a level line, and will lie in the 
plane of the horizon. From also draw OF" parallel to CF', the 
earth's axis. OF" and CF', being parallel, will both be directed 

The "astro- 

Fig. 9. —Relation of Latitude to the 
Elevation of the Pole 

Position of 
the pole in 
the sky. 

The altitude 
of the pole 
equals the 


to their " vanishing point " in the celestial sphere (Sec. 7), 
which is the celestial pole. The angle H' OP" is therefore the 
altitude of the pole as seen at ; and it obviously equals ONQ. 
This fundamental relation, that the altitude of the pole is identical 
with the observer's latitude, cannot be too strongly emphasized. 
Aspect of 33. The Right Sphere. — If the observer is situated at the 

tie eavens gg^j.^]^'j^ CQuator, that is, in latitude zero, the pole will be in his 

as seen irom i 7 ? •> r 

the earth's horizon and the celestial equator will be a vertical circle, coin- 
equator, ciding with the prime vertical (Sec. 14). All heavenly bodies 
will rise and set vertically, and their diurnal circles will all be 
bisected by the horizon, so tliat they will be twelve hours above 
and twelve hours below it ; and the length of the night will 
always equal that of the day (neglecting refraction, Sec. 82). 
This aspect of the heavens is called the right sphere. 

It is worth noting""' that for an observer exactly at the north pole the 
definitions of meridian and azimuth break down, since at that point the 
zenith coincides with the pole. Facing which direction he will, he is 
still looking directly south. If he change his place a few steps, how- 
ever, his zenith will move, and everything will become definite again. 

Aspect of 34. The Parallel Sphere. — If the observer is at the pole of 

the heavens ^^^ ^^^.^j^ ^^^^^ j^-^ latitude is 90°, the cclestial pole will be 

as seen from _ ^ ..... 

the pole. at his zenith and the equator will coincide with the horizon. 
If at the north pole, all the stars north of the celestial equator 
will remain permanently above the horizon, never rising nor 
falling, but sailing around the sky on almucantars, or parallels 
of altitude. The stars in the southern hemisphere, on the 
other hand, will never rise to view. 

Since the sun and moon move among the stars in such a 

way that during half of the time they are north of the equator 

and half the time south of it, they will be half the time above 

The six the horizon and half the time below it, at least approximately, 

months' day ^\^qq ^j^jg statement needs to be slig-htly modified to allow for the 

at the pole. . 

effect of refraction. The moon will be visible for about a fort- 
night each month and the sun for about six months each year. 


35. The Oblique Sphere. — At any station between the poles 
and the equator the pole will be elevated above the horizon, 
and the stars will rise and set in oblique circles^ as shown in 
Fig. 10. Those whose distance from the elevated pole is less 
than PN (the latitude of the observer) will of course never set, 
remaining perpetually visible. The radius of this circle of per- 
petual apparition^ as it is called (the shaded cap around F in 
the figure), is obviously just equal to the height of the pole, 
becoming larger as the latitude increases. On the other hand, 
stars within the same distance 
of the depressed pole will lie 
in the circle of perpetual occul- 
tation, and will never rise 
above the horizon. A star 
exactly on the celestial equa- 
tor will have its diurnal circle 
bisected by the horizon and 
will be above the horizon 
twelve hours. A star north 
of the equator, if the north 
pole is the elevated one, will 
have more than half its diur- 
nal circle above the horizon 
and will be visible for more than twelve hours each day; as, 
for instance, a star at J, rising at B and setting at B'. 

Whenever the sun is north of the celestial equator, the day 
will therefore be longer than the night for all stations in north- 
ern latitude ; how much longer will depend on the latitude of 
the place and the sun's distance from the equator, i.e., its 

36. The Midnight Sun. — If the latitude of the observer is 
such that PJSf in the figure is greater than the sun's polar 
distance or codeclination at the time when the sun is far- 
thest north (about 66^°), the sun will come into the circle of 

Asiiect of 
the heavens 
as seen from 

The circles 
of perpetual 
and occulta- 

Fig. 10. — The Oblique Sphere 

The mid- 
night sun. 



When the 
sun shines 
into north 

perpetual apparition and will make a complete circuit of the 
heavens without setting, until its polar distance again becomes 
less than FN. This happens near the summer solstice at the 
North Cape and at all stations within the Aretie circle. 

Whenever the sun is north of the equator it will in all north 
latitudes rise at a point north of east, as B in the figure, and 
will continue to shine upon every vertical surface that faces 
the north, until, as it ascends, it crosses the prime vertical 
EZW at some point V. 

In the latitude of New York, the sun on the longest days of summer 
is south of the prime vertical only about eight hours of the whole fifteen 
during which it is above the horizon. During seven hours of the day it 
shines into north windows. 

The celestial 

Its horizon 
and circles 
upon it. 

The merid- 
ian ring, its 
and clamp. 

A celestial globe will be of great assistance in studying 
these diurnal phenomena. B}^ means of this it can at once 
be seen what stars never set, which ones never rise, and 
during what part of the twenty-four hours a heavenly body 
at a known declination is above or below the horizon. 

37. The Celestial Globe. — The celestial globe is a ball, usually of 
papier-mache, upon which are drawn the circles of the celestial sphere 
and a map of the stars. It is mounted in a framework which represents 
the horizon and the meridian, in the manner shown by Fig. 11. 

The liorizon, HH' in the figure, is usually a wooden ring three or four 
inches wide, directly supported by the pedestal. It carries upon its upper 
surface at the inner edge a circle marked with degrees for measuring the 
azimuth of any heavenly body, and outside this the so-called "zodiacal 
circles," which give the sun's longitude and the equation of time (Sees. 99 
and 174) for every day of the year. 

The meridian ring, MM', is a circular ring of metal which carries the 
bearings of the axis on w^hich the globe revolves. Things are so arranged, 
or ought to be, that the mathematical axis of the globe is exactly in the 
same i^lane as the graduated face of the ring, which is divided into degrees 
and fractions of a degree, with zero at the equator. The meridian ring 
fits into two notches in the horizon circle and is held underneath the. globe 

prp:limixary considerations and definitions 29 

by a support with a clamp, which enables us to fix it securely in any 
desired position, the mathematical center of the globe being precisely in 
the planes both of the meridian ring and the horizon. 

The hour index on the globe here figured is a pointer like the hour-hand 
of a clock, so attached to the meridian ring at the pole that it can be 
turned around the axis with stiffish friction, but will retain its position 
unchanged when the globe is made to turn under it. It points out the 
time on a small time-circle graduated usually to hours and quarters priiited 
on the surface of the globe. 

The surface of the globe is marked first with the celestial equator (Sec. 20), 
next with the ecliptic (Sec. 23), crossing the equator at an angle of 
23i° (at X in the figure), and 
each of these circles is divided 
into degrees and fractions. 
The equinoctial and solstitial 
colures (Sec. 23) are also al- 
ways represented. As to the 
other circles, usage differs. 
The ordinary way at present 
is to mark the globe with 
twenty-four hour-circles, fif- 
teen degrees apart (the colures 
being two of them), and with 
parallels of declination ten 
degrees apart. 

On the surface of the globe 
are plotted the positions of the 
stars and the outlines of the 

38. To rectify a globe, — 
that is, to set it so as to 
show the aspect of the heavens at any given time, — 

(1) Elevate the north pole of the globe to an angle equal to the 
observer's latitude by means of the graduation on the meridian ring, and 
clamp the ring securely. 

(2) Look up the day of the month on the horizon of the globe and 
opposite to the day find, on the longitude circle, the sun's longitude for 
that day. 

(3) On the ecliptic (on the surface of the globe) find the degree of longi- 
tude thus indicated and bring it to the graduated face of the meridian ring. 

Fig. 11. — The Celestial Globe 

The hour 

drawn on 
surface of 

Setting for 
latitude of 



Setting for 
day of the 

Setting for 
hour of the 

The globe is then set to corres]Dond to (apparent) noon of the day in 
question. (It may be well to mark the place of the sun temporarily with 
a bit of moist paper applied at the proper place in the ecliptic ; it can 
easily be wiped off: after using.) 

(4) Holding the globe fast, so as to keep the place of the sun on the 
meridian, turn the hour index until it shows on the graduated time-circle the 
local mean time of apparent noon, i.e., 12^ ± the equation of time given for 
the day on the horizon ring. (If standard time is used, the hour index must 
be set to the standard time of apparent noon.) 

(5) Finally, turn the globe until the hour for which it is to be set is 
brought to the meridian, as indicated on the hour index. The globe will 
then show the true aspect of the heavens. 

The positions of the moon and planets are not given by this operation, 
since they have no fixed places in the sky and therefore cannot be put upon 
the globe by the maker. If one wants them represented, he must look up 
their right ascensions and declinations for the day in some almanac and 
mark the places on the globe with bits of wax or paper. 


1. What point in the celestial sphere has both its right ascension and 
declination zero ? 

2. What are the celestial latitude tod longitude of this point? 

3. What are the hour angle and azimuth of the zenith? 

4. At what points does the celestial equator cut the horizon ? 

5. What angle does the celestial equator make with the horizon at 
these points, as seen by an observer in latitude 40°? 

6. What if his latitude is 10°? 20°? 50°? 60°? 

7. When the vernal equinox °f is rising on the eastern horizon, what 
angle does the ecliptic make with the horizon at that point for an observer 
in latitude 40° ? 

8. What angle when setting ? 

9. What is the angle between the ecliptic and horizon when the 
autumnal equinox is rising, and when setting ? 

10. Name the fourteen principal points on the celestial sphere (zenith, 
poles, equinoxes, etc.). 

11. W^hat important circles on the celestial sphere have no correlatives 
on the surface of the earth ? 


12. What are the approximate right ascension and declination (a and S) 
of the sun on March 21 and September 22 ? 

13. What is the sun's altitude at noon on March 21 for an observer in 
latitude 42° ? 

14. How far is the sun from the zenith at noon on March 21, as seen 
at Pulkowa, latitude 60°? How far at noon on June 21 ? 

15. On March 21, one hour after sunset, whereabouts in the sky would 
be a star having a right ascension of 7 hours and declination of 40°, the 
observer being in latitude 40° ? 

16. If a star rises to-night at 10 o'clock, at what time (approximately) 
will it rise 30 days hence ? 

17. When the right ascension of the sun is 6 hours, what are its longi- 
tude (A.) and latitude (ft) ? 

18. What, when its a is 12 hours? 

19. What are the latitude and longitude of the north pole of rotation ? 

20. What are the right ascension and declination of the north pole of 
the ecliptic ? 

Note. — None of the above exercises require any calculation beyond a simple 
addition or subtraction. 

21. What are the longitude and declination of the sun when its right 
ascension is 3 hours ? 

^^^^ ( Long. = 47° 27' 59^ 
1 Dec. = 17° 03' 08". 

Note. — This requires the solution of the spherical right angle triangle, in which 
the base is the given a (=45°), the angle adjacent is e (23° 27'), and the parts to be 
found are the hypotenuse X and the other leg opposite e, which is 5. 


Telescopes, and their Accessories and Mountings — Timekeepers and Chronographs 

— The Transit-Instrument — The Prime Vertical Instrument — The Almucantar 

— The Meridian-Circle and Universal Instrument — The Micrometer — The 
Heliometer — The Sextant 

39. Astronomical observations are of various kinds : some- 
times we desire to ascertain the apparent distance between two 
bodies ; sometimes the position which the body occupies at a 
given time, or the time at which it arrives at a given circle of 
the sky, — usually the meridian. Sometimes we wish merely 
to examine its surface, to measure its light, or to investigate 
its spectrum; and for all these purposes special instruments 
have been devised. We propose in this chapter to describe a 
few of the most important at present in use. 

40. Telescopes in General. — Telescopes are of two kinds, 
refracting and reflecting. The former were first invented 
and are much more used, but the largest instruments which 

Fundamental have cvcr been made are reflectors. In both the fundamental 
prmcipieof principle is identical. The larsfe lens, or mirror, — the "obiect- 

the telescope. ^ ^ . . . 

ive " of the instrument — forms at its focus a "real" image 

of the object looked at, and this image is then examined 

and magnified by the eyepiece, which in principle is only a 


Essential 41. The Simple Refracting Telescope. — This consists essen- 

^i^™^^f !^ *^^ tially, as shown in Fig. 12, of two convex lenses, one the object- 

ing tele- glass A, of large size and long focus ; the other, the eyepiece 

scope. ^^ Qf short focus ; the two being set at a distance nearly equal to 

the sum of their focal lengths. Recalling the optical principles 



of the formation of images by lenses,^ we see that if the 

instrument is pointed toward the moon, for instance, all the 

rays that strike the object-glass from the top of the object will 

come to a focus at a, while those from the bottom will come to 

a focus at J, and similarly with rays from other points on the 

surface of the moon. We shall therefore get in the " focal Real image 

plane" of the obi'ect-p-lass a small inverted "real" ima^e of the ?_\^^t^ ^^ 

r J t> o object-glass, 

moon, so that if a photographic plate is inserted in the focal 
plane at ab and properly exposed, we shall get a picture of the 

The size of the picture will depend upon the apparent Size of the 
angular diameter of the object and the distance of the image "^^^e- 
ab from the object-glass, and is determined by the condition 



— -^^^^ 

Fig. 12. — The Simple Refracting Telescope 

that, a8 Been from point (the optical center of the object-glass), 
the object and its image subtend equal angles, since rays which 
pass through the point suffer no sensible deviation. 

, If the focal length of the lens y1 is 10 feet, then the image of the moon 
formed by it will appear, when viewed from a distance of 1 feet, just as 
large as the moon itself ; from a distance of 1 foot, the image will, of 
course, appear ten times as large. 

With such an object-glass, therefore, even without an eyepiece, one 
can see the mountains of the moon and satellites of Jupiter by simply 
putting the eye in the line of the rays, at a distance of 10 or 12 inches 
back of the eyepiece hole, the eyepiece itself having been, of course, 

1 In this explanation we use the approximate theory of lenses (in which 
their thickness is neglected), as given in the elementary text-books. The more 
exact theory would require some slight modification in statements, but none of 
substantial importance. 


42. Magnifying Power. — If we use the naked eye, one 
cannot, unless near-sighted, see the image distinctly from a 
di&tance much less than 10 inches; but if we use a magnifying- 
lens of 1-inch focus, we can view it from a distance of only 
an inch, and it will look correspondingly larger. Without 
stopping to demonstrate the principle, the magnifying power is 
simply equal to the quotient obtained by dividing the focal length 
of the object-glass by that of the eye-lens ; or, as a formula. 

Formula for 

themagni- M = F/f\ that is, Od/cd in the figure. 

fying power. 

If, for ex-ample, the focal length of the object-glass be 4 feet 
and that of the eye-lens one quarter of an inch, then 

jf=48 ^1 = 4x48 = 192. 

A magnifying power of unity, however, is often spoken of as " no magni- 
fying power at all," since the image appears of the same size as the object. 

The magnifying power of the telescope is changed at pleasure by simply 
changing the eyepiece (see Sec. 47). 

Light-gath- 43. Llght-Gathcrlng Power of the Telescope and Brightness 
ering power of the Image. — This depends not upon the focal length of the 
to the square object-glass, but upou its diameter; or, more strictly, its area. 
of the diam- If we estimate the diameter of the pupil of the eye at one fifth 
of an inch, then (neglecting the loss in transmission through 
the lenses) a telescope 1 inch in diameter collects into the 
image of a star twenty-five times as much light as the naked 
eye receives ; and the great Yerkes telescope of 40 inches 
in diameter gathers 40000 times as much, or about 35000 
after allowing for the losses. The amount of light collected is 
proportional to the square of the diameter of the object-glass. 

The apparent brightness of an object which, like the moon or 
a planet, shows a disk, is not, however, increased in any such 
ratio, because the light gathered by the object-glass is spread 
out by the magnifying power of the eyepiece. In fact, it can be 
demonstrated that no optical arrangement whatever can show 

eter of the 



an extended surface brighter than it appears to the naked eye. 
But the total quantity of Ught in the image of the object greatly 
exceeds that which is available for vision with the naked eye, 
and objects which, like the stars, are mere luminous points, 
have their brightness immensely increased, so that with the tele- 
scope millions otherwise invisible are brought to light. With the 
telescope, also, the brighter stars are easily seen in the daytime. 

44. The Achromatic Telescope. — A single lens cannot bring 
the rays which emanate from a single point in the object to any 
exact focus, since the rays of different color (wave-length) are 
differently refracted, the blue more than the green, and this 
more than the red. In consequence of this so-called " chromatic 
aberration," the simple refract- 
ing telescope is a very poor 

About 1760 it was discov- 
ered in England that by making 
the object-glass of two or more 
lenses of different kinds of glass the chromatic aberration can 
be nearly corrected. Object-glasses so made — no others are 
now in common use — are called achromatic^ and they fulfil 
with reasonable approximation, though not perfectly, the con- 
dition of distinctness ; namely, that the rays which emanate 
from any single point in the object should be collected to a 
single point in the image. In practice, only two lenses are 
ordinarily used in the construction of an astronomical object- 
glass, — a convex of crown-glass, and a concave of flint-glass, 
the curves of the two lenses and the distances between them 
being so chosen as to give the best possible correction of the 

No optical 
ment can 
increase the 
of an 

of a single- 
lens object- 

Garm -^^"^^^ 


Fig. 13. — Different Forms of the 
Achromatic Object-Glass 

The achro- 
matic lens. 

1 By making the telescope extremely long in proportion to its diameter, the 
distinctness of the image is considerably improved, and in the middle of the 
seventeenth century instruments more than 200 feet in length were used by 
Cassini and others. Saturn's rings and several of his satellites were discovered 
by Huyghens and Cassini with instruments of this kind. 



tism of 

More perfect 
lenses from 
new kinds 
of glass. 

The spuri- 
ous disk of 
a star. 

spherical aberration as well as of the chromatic. Many forms 
of object-glass are made, three of which are shown in Fig. 13. 

45. Secondary Spectrum. — It is not possible to obtain a 
perfect correction of color with the only kinds of glass which 
were available until very recently. Ordinary achromatic lenses, 
even the best of them, show around every bright object a strong 
purple halo, due to red and blue rays which are both brought 
to a focus further from the object-glass than are the yellow and 
green. This halo seriously injures the definition and makes it 
difficult to see small stars very near a bright one. It is specially 
obnoxious in large instruments. 

Much is hoped from the new varieties of glass now being 
made at Jena in Germany. Several telescopes of considerable 
size have already been constructed, of which the lenses are 
practically aplanatic ; that is, sensibly free from both spherical 
and chromatic aberration. Possibly a new era in telescope 
making is opening with the new century. 

46. Diffraction and Spurious Disks. — Even if a lens were 
absolutely perfect as regards the correction of aberrations, 
it would still be unable to fulfil strictly the condition of 

Since light consists of waves of finite length, the image of a 
luminous point can never be also a point, but necessarily, on 
account of " diffraction," consists of a central disk of finite 
diameter, surrounded by a series of " interference " rings ; and 
the image of a line is a streak and not a line. The diameter 
of the " spurious disk " of a star, as it is called, varies inversely 
with the diameter of the ohject-glass ; the larger the telescope, the 
smaller the image of a star with a given magnifying power. 

With a good 41^-inch telescope and a power of about 120, the 
image of a small star, when the air is perfectly steady (which 
unfortunately seldom happens), is a clean, round disk, about 1" 
in diameter, with a bright ring around it, separated from the 
disk by a dark space about as wide as the disk. With a 9-inch 

Formula for 
diameter of 


instrument the disk has a diameter of 0''.5, — just half as great; 
with the Yerkes telescope, about 0".ll. The angular diameter 
of a star disk in a telescope the aperture of which is a inches 
is, therefore, given by the following formula, due to Dawes : 

4" ^ 
d" = ^^- 

flf spurious 


If the magnifying power is too great (more than about sixty 
to the inch of aperture), the disk of a star will become ill-defined 
at the edge ; so that there is very little use with most objects in 
pushing the magnifying power any higher. 

This effect of "idiffraction " has much to do with the supe- 
riority of large instruments in showing minute details ; no 
increase of magnifying power on a small telescope can exhibit Superiority 
the object as sharply as the same power on a large one, pro- ^^^^^^s® 
vided, of course, that the object-glasses are equally good in work- glasses in 
manship and that the atmospheric conditions are satisfactory, f^efining 

T-» • /» 7 • T 7 • • 7 power. 

(But a given amount oj atmospheric disturbance injures the per- 
formance of a large telescope much more than that of a small one.) 

47. Eyepieces, or << Oculars. ^^ — For some purposes the simple 
convex lens is the best eyepiece possible ; but it performs well 
only for a small object, like a close double star, exactly in the 
center of the field of view. Generally, therefore, we employ 
eyepieces composed of two or more lenses, which give a larger 
field of view than a single lens and define fairly well over the 
whole extent of the field. They fall into two general classes, 
the positive and the negative. 

The positive eyepieces are much more generally useful. They Positive 
act as simple magnifying-glasses and can be taken out of th%^^®P^®^®^' 
telescope and used as hand magnifiers if desired. The image of 
the object formed by the object-glass lies outside of this kind 
of eyepiece, between it and the object-glass. 

In the negative eyepieces, on the other hand, the rays from Negative 
the object are intercepted by the so-called " field lens " before eyepieces. 



reaching the focus, and the image is formed inside the eye- 
piece. It cannot therefore be used as a hand magnifier. 

Fig. 14 shows the two most usual forms of eyepiece, and 
also the "- solid eyepiece " constructed by Steinheil ; but there 
are a multitude of various kinds. All these eyepieces show 
the object inverted, which is of no importance in astronomical 

steinheil 'Monocentric' 






Fig. 14. — Various Forms of Telescope Eyepiece 

It is evident that in an achromatic telescope the object-glass is by far 
the most important and expensive member of the instrument. It costs, 
according to size, from ^100 up to $65000, while the eyepieces cost only 
from $2 to $26 apiece, and every telescope of any pretension possesses a 
considerable stock, of various magnifying powers. 

48. Reticle. — If the telescope is to be used for pointing 
The reticle, upon an object, it must be provided with a "reticle" of some 
sort. The simplest is a frame with two spider-lines stretched 
across it at right angles to each other, their intersection being 
the point of reference. This reticle is placed, not at or near 
the object-glass, as often supposed, but in the focal plane^ as 
ah in Fig. 12 (Sec. 41). Of course, positive eyepieces only can 
be used in connection with such a reticle, though in sextant 
telescopes a negative eyepiece is sometimes used with a pair 
of cross-wires placed between the two lenses of the eyepiece. 
Sometimes a glass plate with fine lines ruled upon it is used 
instead of spider-lines. In order to make the lines of the 


reticle visible at night, a faint light is reflected into the instru- 
ment by some one of various arrangements devised for the 

49. The Reflecting Telescope. — About 16T0, when the chro- The reflect- 
matic aberration of refractors first came to be understood (in ^^^ 

^ scope. 

consequence of Newton's discovery of the decomposition of 
light), the reflecting telescope was invented. For nearly one 
hundred and fifty years it held its place as the chief instru- 
ment for star-gazing. There are several varieties, differing in 
the Avay in which the image formed by the mirror is brought 




* Eye 



^ — > \i[- 

Fig. 15. — Kefleetiug Telescopes 

within reach of the magnifying eyepiece. Fig. 15 illustrates 

three of the most common forms. The Newtonian is most A^arious 

used, but one or two larg^e instruments are of the Casses^rainian ^*^^'™^ ^^ *^® 

^ _ '^ reflector. 

form, which is exactly like the Gregorian shown in the figure 
(now almost obsolete), with the exception that the small mirror 
is convex instead of concave. 

In the Herschelian, or "front view" form, the large mirror 
is slightly inclined, throwing the rays to the edge of the open 
end of the tube, so that the secondary mirror is dispensed with, 
and the observer stands with his back to the object. This is 
practicable only with very large instruments, since the head 



Mirrors of 
silver on 

of the 
over the 

of the 


of the observer partly obstructs the light; the image also is 
somewhat distorted, and at present this construction is never 

Until about 1870, the large mirror (technically speculum) 
was alw^ays made of speculum-metal, a composition of copper 
and tin. It is now usually made of glass, silvered on the front 
surface by a chemical process. When new, these silvered films 
reflect much more light than the old speculum-metal ; they tar- 
nish rather easily, but fortunately can be easily renewed. 

50. Relative Advantages of Reflectors and Refractors. — There 
is much earnest discussion on this point, each form of instru- 
ment having its earnest partisans. On the whole, however, 
the refractor is usually better. Up to a certain limit, never 
yet reached, it gives more light than a reflector of the same 
size, defines better under all ordinary conditions, has a wider 
field of view, is more manageable and convenient, and more 
permanent; the speculum of a reflector usually needs to be 
resilvered every few years, while a carefully used object-glass 
never deteriorates. 

The reflector is of course far less expensive than a refractor 
of the same size, and its absolute achromatism is a great advan- 
tage in certain lines of work, photographic and spectroscopic. 

For a fuller discussion of the matter, see General Astro7iomy. 

51. Large Telescopes. — The largest refractors^ at present (1901) exist- 
ing are those of the Yerkes Observatory (40 inches in diameter and 65 
feet long), and the telescope of the Lick Observatory, which has an 
aperture of 36 inches and a focal length of 56 feet. There are about 
fourteen others which have apertures not less than 2 feet. The object 
lenses of more than half of these instruments, including both of the 
largest, were made (that is, ground and figured) in this country by the 
Clarks of Cambridgeport. The glass itself was made by various firms 
m Europe. 

1 No account is taken in this reckoning of the great 48-inch telescope of the 
Paris Exposition. It is not certain as yet how it will turn out from an astro- 
nomical point of view. 



The frontispiece is the great Potsdam double telescope, — two mounted 
together, — one 31^ inches in diameter for photography, the other 20 inches 
in diameter for visual observations ; the focal length of both is about 43 
feet. It was erected in 1899. 

At the head of the reflectors stands the enormous instrument of Lord^ 
Rosse of Birr Castle, 6 feet in diameter and 60 feet long, made in 
1842, and still used occasionally. Next in size, but probably superior Large 
in power, comes the 5-foot silver- on-glass reflector of Mr. Common, at reflectors. 
Ealing, England, completed in 1889 ; and then follow a number (four or 
five) of 4-foot telescopes, — that of Herschel (erected in 1789, but long .^j.^ 

ago dismantled) being the first, while the 
instrument at Paris is the only one of this 
size now in active use. 

In this country the only large reflector 
actually at work is the 3-foot instrument at 
the Lick Observatory (made by Mr. Common 
and presented to the observatory by Mr. 
Crossley), with which Keeler made his won- 
derful photographs of nebulae, some of which 
are figured in the last chapter of this book. 
Another of 2-feet aperture has just been 
mounted at the Yerkes Observatory. 

52. Mounting of a Telescope. — A 

telescope, however excellent optically, 
is of little scientific use unless firmly 
and conveniently mounted.^ 

At present nearly all but small 
portable instruments are mounted as Equatorials. Fig. 16 rep- 
resents the arrangement schematically. Its essential feature is 
that the "principal axis " — the one which moves in fixed bear- 
ings attached to the pier and is called the polar axis — is inclined 
so as to point towards the celestial pole. The graduated circle 
H attached to it is therefore parallel to the celestial equator, 

Fig. 16. — The Equatorial 




1 We may add that it must be mounted where it can be pointed directly at 
the stars, without any intervening window-glass between it and the object. We 
have known purchasers of telescopes to complain bitterly because their instru- 
ments would not show Saturn well through a closed window. 



of equa- 

Permits use 
of clock- 

Makes it 
easy to find 
objects too 
faint to be 

Use of 
in determin- 
ing position 
of planets or 

and is usually called the hour-circle of the instrument, — 
sometimes the right-ascension circle. At the upper extremity 
of the polar axis a sleeve is fastened, which carries the 
declination axis D passing through it. To one end of this 
declination axis is attached the telescope tube T, and at the 
other end the declination circle (7, and a counterpoise if 

53. The advantages of the equatorial mounting are very 
great. In the first place, when the telescope is once pointed 
upon an object it is not necessary to turn the declination 
axis at all in order to keep the object in view, but only to 
turn the polar axis with a perfectly uniform motion, which 
can be, and usually is, given by clockwork (not shown in the 

In the next place, it is very easy to find an object, even if 
invisible to the eye (like a faint comet, or a star in the day- 
time), provided we know its right ascension and declination 
and have the sidereal time, — a sidereal clock or chronometer 
being an indispensable accessory of the equatorial. We set 
the declination circle by its vernier to the declination of the 
object and then turn the polar axis until the hour-circle shows 
the proper hour angle., which is simply the difference between 
the right ascension of the object and the sidereal time at the 
moment. When the telescope has been so set the object will 
be found in the field of view, provided a low-power eyepiece is 
used. On account of refraction the setting does not direct 
the instrument precisely to the apparent place of the object, 
but only very near it. 

The equatorial does not give very accurate positions of 
heavenly bodies by means of the direct readings of its circles, 
but it can be used as explained later in Sec. 117 to determine 
with great precision the difference between the position of a 
known star and that of a comet or planet; and this answers 
the purpose as well as a direct determination. 



The frontispiece shows the equatorial mounting of the great Potsdam tel- 
escope. Fig. 173 (Sec. 536) represents another form of equatorial mounting, 
adopted for several of the instruments of the photographic campaign. Lord 
Rosse's great reflector is not mounted equatorially, nor was Herschel's 4-foot 
reflector, but nearly all the other reflectors referred to above are equatorials. 

54. Other Mountings. — With very large telescopes this 
mounting becomes unwieldy, notwithstanding the ingenious 
electrical and other arrangements by which the observer at 

Fig. 17. — The Equatorial Coude. 

the eyepiece is enabled to control its motions. The enor- 
mous rotating dome — that of the Yerkes Observatory is 90 
feet in diameter — and the requisite elevating floor are also 
extremely expensive, so that at present there is among astron- 
omers a tendency to adopt plans by which the telescope may 
be fixed in its position, while the light is brought to the eye- 
piece by one or more reflections from plane mirrors. 

Fig. 17 represents the smaller equatorial coude, or '< elbowed equato- 
rial," of the Paris Observatory. A silvered mirror at an angle of 45° in 









the box in front of the object-glass, and another one in the cube at the 
center of the instrument, effect the necessary changes in the direction of the 
ray. The observer sits motionless, under cover, at the eyepiece, looking 
downward towards the south, at an angle equal to the latitude of the place. 
A much larger, similar instrument, since mounted at the same observatory, 
has an aperture of 24 inches and a focal length of about 60 feet. Three or 
'fH9ur instruments of this sort are now in use. 

' Another arrangement is to place the telescope horizontally, pointing 
towards the south, and to direct the light from the object into it by 
reflection from the mirror of a so-called siderostat. This is a simple 
plane mirror larger than the object-glass, properly mounted and driven 
by clockwork so as to send the reflected rays horizontally always in the 
same direction, and having connections by which its motions can be con- 
trolled from the eye end of the telescope. The great telescope of the Paris 
Exposition of 1900 was arranged in this way. 

The eoelostat is a slightly different arrangement, in which the plane 
mirror, mounted upon a polar axis, revolves at half the diurnal rate, and 
the telescope, while retaining one fixed position for a body in a given 
declination, has to change its position to observe bodies in a diiferent 
declination. There are still other forms in which a large reflector is used 
to give the rays a convenient direction. 

But the use of the mirror or mirrors involves considerable loss of light ; 
and what is worse, if the mirror is large it is extremely difficult to figure 
the surface with the requisite accuracy, and to prevent slight distortions 
by variations of temperature and changes of position. As a consequence, 
definition is seldom as satisfactory as with telescopes pointed directly to 
the heavens ; still, in certain operations of astronomical photography, the 
siderostat and eoelostat are extremely useful. 

55. Timekeepers and Recorders Obviously a good clock 

or chronometer is an essential instrument of the observatory. 
Importance The invention of the pendulum clock by Huyghens in 1657 
was almost as important to the advancement of astronomy as 
that of the telescope by Galileo ; and the improvement of the 
clock and chronometer through the invention of temperature 
compensation by Harrison and Graham in the eighteenth cen- 
tury is fully comparable with the improvement of the telescope 
by the achromatic object-glass. 

of Huy- 
of the 




The astronomical clock differs from any other clock only in The astro- 
being made with extreme care and in having a pendulum so "^^"^^^'^^ 
constructed that its rate will not be sensibly affected by 
changes of temperature. The mercurial pen- 
dulum is most common, but other forms are 
also used. (See Fig. 18.) 

The pendulum usually beats seconds (rarely 
half seconds), and the clock face ordinarily has 
its second-hand, minute-hand, and hour-hand 
each moving on a separate center, the hour- 
hand making its revolution not in twelve 
hours, as in an ordinary clock, but in twenty- 
four, the hours being numbered accordingly. 

In cases where the extremest accuracy of 
performance is required, the clock is placed 
in an underground chamber, where the tem- 
perature varies only slightly or not at all, and 
is besides inclosed in an air-tight case, within 
which the air is kept at a uniform pressure, 
since changes in the density of the air slightly 
affect the swing of the pendulum. Usually a 
clock loses about one quarter of a second a 
day for a rise of one inch in the barometer. 

Finally, also, the astronomical clock is usu- 
ally fitted with some arrane^ement for makinsf i- Graham's Pendulum 

T , . . . . 2. Zinc-steel Pendulum 

or breaking an electric circuit at every second or 

every other second, so that its beats can be communicated tele- The break- 
graphically to all parts of the observatory. The minute is usually ^^^■^^^^• 
marked either by the omission of a second or by a double tick. 

56. Error and Rate. — The error or correction of a clock is the Error and 
amount which must be added {algehraieally) to its face indication ^^ ^' 
to give the true time ; -\- when slow, — when fast. The rate is 
the amount it loses or gains daily ; -f when losing, — when 
gaining. Sometimes the hourly rate is given instead of the 

Fig. 18 
Compensation Pen- 



The chro- 

of time 
by eye and 

daily. The error is adjusted by simply setting the hands ; the 
rate by raising or lowering the pendulum bob, or for delicate 
final adjustment without stopping the clock, by adding or 
removing small pieces of metal on the cover of the cylindrical 
vessel which usually constitutes the pendulum bob. 

Perfection in an astronomical clock consists in its maintain- 
ing a constant rate^ i.e., in gaining or losing precisely the same 
amount each day; for convenience the rate should be small, 
and is usually kept less than half a second daily. But this is 
a mere matter of adjustment. 

57. The Chronometer. — The pendulum clock not being port- 
able, it is necessary to provide timekeepers that are so. The 
chronometer is merely a carefully made watch with a balance- 
wheel compensated to run, as nearly as possible, at the same 
rate in different temperatures, and with a peculiar escapement, 
which, though unsuited to ordinary usage, gives better results 
than any other when treated carefully. 

The box chronometer used on shipboard is usually about 
twice the diameter of a common pocket watch, and is mounted 
gimbals " so as to remain horizontal at all times, notwith- 


standing the motion of the vessel. It usually beats half seconds. 

It is not possible to secure in the chronometer balance as 
perfect a temperature correction as in the pendulum, and for 
this and other reasons the best chronometers cannot quite com- 
pete with the best clocks in precision; but they are sufficiently 
accurate for most purposes, and of course are vastly more con- 
venient for field operations, while at sea they are simply indis- 
pensable. Never turn the hands of a chronometer hachward ; it 
may ruin the escapement. 

58. Eye-and-Ear Method of Observation. — The old-fashioned 
method of time observation consisted simply in noting by " eye 
and ear " the moment (in seconds and tenths of a second) when 
the phenomenon occurred ; as, for instance, when a star passed 
some wire of the reticle. The tenths, of course, are merely 



estimated, but the skilful observer seldom errs by a whole 
tenth in his estimation. Skill and accuracy in this method are 
acquired only by long practice. 

59. Telegraphic Method ; the Chronograph. — At present observation 
such observations are usually made by the help of electricity. ^^ means of 

the chrono- 

50 s. 


n r^r\r\nnnr\r\nnr\nf\r\n 

9 h. 35 m. 00.0 s. 





Fig. 19. — A Chronograph Record 

Fig. 20. — A Chronograph 
By Warner & Swasey 

The clock is so arranged that at every beat (or every other 
beat) of the pendulum an electric circuit is made or broken 
for an instant, and this causes a sudden sideways jerk in the 


armature of an electromagnet, like that of a telegraph sounder. 
This armature carries a fountain-pen, which writes upon a 
sheet of paper wrapped around a cylinder six or seven inches 
in diameter, which cylinder itself is turned uniformly by clock- 
work once a minute ; at the same time the pen carriage is drawn 
slowly along, so that the marks on the paper form a continuous 
helix, graduated into second or two-second spaces by the clock 
beats. When taken from the cylinder, the paper presents the 
appearance of an ordinary page crossed by parallel lines spaced 
off into two-second lengths, as shown in Fig. 19, which is part 
of an actual record. 

Fig. 20 represents a chronograph of the usual American form. 

The observer, at the moment when a star crosses the wire, 
presses a " key " which he holds in his hand, and thus inter- 
polates a mark of his own among the clock beats on the sheet ; 
as, for instance, at X and Y in the figure. Since the beginning 
of each minute is indicated on the sheet in some way by the 
mechanism which produces the clock beats, it is very easy to 
read the time of X and Y by applying a suitable scale, the 
beginning of the mark made by the key being the moment of 

In the figure the initial minute marked when the chronograph was 
started happened to be 9^35°^, the zero in the case of this clock being- 
indicated by a double beat. The signal at X, therefore, was made at 
9^35^558.45, and that of Y at 9^36^588.63. The "rattle" just pre- 
ceding X was the signal that a star was approaching the transit wire. 
In European observatories the record is usually made by a more simple 
but less convenient apparatus upon a long fillet or ribbon of paper drawn 
slowly along. At a few observatories in this country a more complicated 
The print- printing chronograph^ invented by Professor Hough of the Dearborn Observa- 
mg clirono- tory, is used. By this the minutes, seconds, and hundredths of a second 
grap 1. ^^^ actually printed upon the fillet in type, like the record of sales on a 

stock telegraph. 

60. Meridian Observations. — A large proportion of all astro- 
nomical observations for determining the positions of the 



heavenly bodies are made when the body is crossing the meridian 
or is very near it. At that time the effects of refraction and 
parallax (to be discussed later) are a minimum, and as they act 
only vertically they do not affect the time when a body crosses 
the meridian nor, consequently, its observed right ascension. 
In any other part of the sky both these coordinates are affected, 
and the calculation of the correction requires the computation of 
the "parallactic angle" in the astronomical triangle (Sec. 31). 

61. The transit-instrument is the instrument used in connec- 
tion with a sidereal clock or chronometer, and often with a 
chronograph, to observe the time 
of a star's transit^ or passage across 
the meridian. If the " error " of 
the sidereal clock at the moment 
is known and allowed for, the 
corrected time of the observation 
will he the right ascension of the 
star (Sec. 26). 

Vice versa^ if the right ascen- 
sion is known, the error of the 
clock will be the difference be- 
tween the right ascension of the 
object and the time observed. 

The instrument (Fig. 21) con- 
sists essentially of a telescope carrying at the eye end a reticle 
and mounted on a stiff axis that turns in V-shaped bearings 
called "Y's," which can have their position adjusted so as to 
make the axis exactly perpendicular to the meridian. A 
delicate spirit-level, which can be placed upon the pivots of 
the axis to measure any slight deviation from horizontality, 
is an essential accessory ; and it is practically necessary to 
have a small graduated circle attached to the instrument, 
in order to set it at the proper elevation for the star which is 
to be observed. 

of observa- 
tions on the 

Fig. 21, — The Transit-Instrument 

The transit- 

The level, 
circle, and 



The transit 

from shaki- 
ness and 
flexure of 

of construc- 

It is desirable, also, that the instrument should have a 
reversing apparatus by which the axis may be easily lifted and 
safely reversed in the Y's without jar or shock. 

The reticle usually contains from five to fifteen " vertical 
wires " crossed by two horizontal ones. Fig. 22 shows the 
reticle of a small transit intended for observations by " eye 
and ear." When the chronograph is to be used, the wires are 
much more numerous and placed nearer together. 

In order to make the wires visible at night the field must be illuminated. 
For this purpose one of the pivots of the instrument is pierced (some- 
times both of them), so that the light from a lamp will shine through the 

axis upon a small reflector placed in the central 
cube of the instrument, where the axis and the 
tube are joined. This sends sufficient light 
towards the eye to illuminate the field, while 
it does not cut off any considerable portion of 
the rays from the object. 

The observation consists in noting 
the instant, as shown by the clock or 
chronometer, in hours, minutes, seconds, 
and tenths of a second, at which the 
star crosses each wire of the reticle. 
62. The instrument must be thoroughly rigid, without any 
loose joints or shakiness, especially in the mounting of the ohject- 
glass and reticle. Moreover, the two pivots should be of the 
same diameter, accurately round, without taper, and precisely 
in line with each other ; in other words, they must be portions 
of one and the same geometrical cylinder. To fulfil this condition 
taxes the highest skill of the mechanician. 

When exactly adjusted, the middle wire of such an instru- 
ment always precisely coincides with the meridian^ however 
the instrument may be turned on its axis; and the sidereal 
time when a star crosses that wire is therefore the star's right 

Fig. 22. —Reticle of the 



Another form of. the instrument now much used is often called the 
broken transit, of which Fig. 23 is a representation. A reflector (usually 
a right-angled prism) in the central cube of the instrument directs the 
rays horizontally through one end of the axis where the eyepiece is 

Fig. 23. — a Broken Transit 
By Warner & Swasey 

placed, so that whatever may be the elevation of the star the observer 
looks straight forward horizontally, without needing to change his position. 
The instrument is very convenient, but is usually subject to rather a large 
error, due to flexure of the axis, which, even if it exists, produces no such 
effect in transits of ordinary form. The error is, however, easily determined 
and allowed for if the axis is not too slender. 




Tests of 

nence of 

63. Adjustments of the Transit. — These are four in number: 

(1) The reticle must be exactly in the focal plane of the 
object-glass and the middle wire accurately vertical. 

(2) The line of collimation (i.e., the line which joins the optical 
center of the object-glass to the middle wire) must be exactly 
perpendicular to the axis of rotation. This may be tested by 
pointing on a distant mark and then reversing the instrument. 
The middle wire must still bisect the mark after the reversal. 
If not, the reticle must be adjusted by the screws provided for 
the purpose. 

(3) The axis must be level. This adjustment is made mechan- 
ically by the help of the spirit-level. One of the Y's has a 
screw by which it can be slightly raised or lowered, as may be 

(4) The azimuth of the axis must be exactly 90°; i.e., the 
axis must point exactly east and west. This adjustment is 
made by means of star observations, with the help of the side- 
real clock. 

Without going into detail, we may say that if the instrument 
is correctly adjusted, the time occupied by a star near the pole, 
in passing from its transit across the middle wire above the 
pole to its next transit below the pole, must be exactly twelve 
sidereal hours. Moreover, if two stars are observed, one near 
the pole and another near the equator, the difference between 
their times of transit ought to be precisely equal to their differ- 
ence of right ascension. By utilizing these principles the 
astronomer can determine the error of azimuth adjustment and 
correct it. 

But it is to be remembered that no adjustments, however 
carefully made, will be absolutely exact or remain permanently 
correct, on account of changes in temperature which affect the 
instrument and the pier on which it is mounted. In cases 
where extreme accuracy of results is required, the slight errors 
which remain after the most careful adjustment must be 


determined from the observations themselves by means of the 
little discrepancies between the results obtained from stars at 
different distances from the pole. The methods to be used are 
taught in practical astronomy. 

64. Personal Equation. — It is found that skilled observers Personal 
are in the habit of noting the passage of a star across the ®^"^*^^"- 
transit wire slightly too late or too early by an amount which 

is different for each observer, but nearly constant for each. 
This is called the observer's personal equation^ and in some 
cases for eye-and-ear observation is as much as half a second. 
In the telegraphic method it is much less, seldom exceeding 
OM. It is an extremely troublesome error, because it varies 
with the nature and brightness of the object and with the 
observer's position and physical condition. 

Various devices have been proposed for dealing with it ; either by Mechanical 

measuring its amount, or by eliminating it by means of some apparatus method of 

which reduces the observation to the accurate bisection of the star disk, ^® ^^^ ^^ 

. . .01 personal 

made to appear to be at rest by a clockwork motion given to the eyepiece, gguation 

and carrying with it a " micrometer wire " which is under the control of 

the observer. When the bisection is satisfactory he touches a key which 

instantly stops the motion and registers the time upon the chronograph ; 

afterwards, at his leisure, he measures the distance of his micrometer wire 

from the central wire of the reticle. In this way the disturbing effect of 

the star's motion is eliminated. 

65. The Photochronograph. — Another method, and one of the most Photo- 
promising, is by means of photography. The eyepiece of the transit is graphic 
removed, and a small photographic plate, about as large as a microscope 
slide, is placed just back of the reticle, so arranged in the frame which 
holds it that it can move up and down slightly under the action of an 
electromagnet connected with the standard-clock circuit. When a star 
impresses its " trail " on the plate, the trail is broken every second (or 
every other second) by the clock, like the marks on a chronograph sheet, 
so that it consists of a row of small dashes. The image of the reticle 
wires is also imprinted upon the plate by holding a small lamp for an 
instant in front of the object-glass. 

During the passage of the star some particular second is marked on 
the plate by cutting off the clock circuit for two or three seconds, or by 

of transits. 



The prime 



Its use. 

The ahnu- 
cantar and 
its use. 

making a rattle, allowing the beats to resume their regular course at some 
instant recorded in the note-book. After the plate is developed, its inspec- 
tion and measurement under a microscope will show at what second and 
fraction of a second the star passed each reticle wire. But this part of the 
operation is laborious. On the other hand, the expensive and troublesome 
chronograph is dispensed wdth. 

66. The Prime Vertical Instrument. — For certain purposes 
a transit-instrument, provided with an apparatus for rapid 
reversal, is turned quarter way round and mounted with its 
axis north and souths so that the plane of rotation lies east and 
west instead of in the meridian. It is then called the " prime 
vertical instrument." It may be used for determining the lati- 
tude of the observer, the precise declination of such stars as 
cross the meridian between the zenith and equator, and any 
minute change due to "aberration" and to slight movements 
of the terrestrial pole. (See Sec. 94.) 

The observation consists in noting the instant when the star 
crosses (obliquely) the middle wire of the reticle. 

67. The Almucantar. — This is an instrument invented about 
1885 by Dr. S. C. Chandler of Cambridge, U.S., for the pur- 
pose of observing the time at which stars cross, not the meridian 
or any vertical circle, but some given parallel of altitude^ usually 
the " alnfucantar " of the pole. From such observations can be 
determined with great accuracy the error of the clock, the decli- 
nation of the stars observed, or the latitude of the observer. 

It consists of a firm base carrying a tank containing mercury, 
on which swims a float which carries the observing telescope, 
its inclination being preserved absolutely constant by the prin- 
ciple of flotation. This dispenses with the necessity of using 
spirit-levels (which are always more or less unsatisfactory) for 
determining the inclination. 

The telescope is sometimes placed horizontally on the float, while a mirror 
in front of its object-glass brings down the rays of the star. Two such 
instruments of considerable size have been built since 1899 and give prom- 
ising results, — one at Cambridge, England, the other at Cleveland, Ohio. 



68. Th*e Meridian-Circle. — This is a transit-instrument of The merid- 
large size and most careful construction, luith the addition of a ^^^-^i^'ce. 

o . essentials of 

large graduated circle attached to the axis and turning luith it. its construc- 

Fig. 24. — Meridian-Circle in United States Naval Observatory, Washington 

By "Warner «& Swasey 

The utmost resources of mechanical art are expended in gradu- 
ating this circle with precision. The divisions are now usually 
made either two minutes or five minutes of arc, and the farther 



Its zero 

tion of the 
polar point. 

tion of the 
nadir point. 

subdivision is effected by so-called " reading microscopes," 
four of which at least are always used in the case of a large 
instrument. (For a description of the reading microscope, the 
reader is referred to General Astronomy^ Art. 64, or to Camp- 
bell's Practical Astronomy.) By means of these microscopes 
the " reading of the circle " is made in degrees, minutes, sec- 
onds, and tenths of a second of arc, the tenths being obtained 
by estimation. 

On a circle 2 feet in diameter 1" of arc is only about t^^oo P^-i't of 
an inch ; an error of that amount is now very seldom made by reputable 
constructors in placing a graduation line, or by a good observer in reading 
the instrument with the microscope. 

Fig. 24 represents the new meridian-circle of the United States Naval 
Observatory at Washington, with a 6-inch telescope and circles about 
27 inches in diameter. 

69. Zero Points. — The instrument is used to measure the 
altitude or else the polar distance of a heavenly body at the 
time when it is crossing the meridian. As a preliminary we 
must determine some zero point upon the circle, — the nadir 
point or horizontal point, if we wish to measure altitudes or 
zenith-distances ; the polar point or equator point, if polar dis- 
tances or declinations. The polar point is determined by taking 
the circle reading for some star near the pole when it crosses 
the meridian above the pole, and then doing the same thing 
again twelve hours later when it crosses it below. The mean 
of the two readings corrected for refraction will be the reading 
which the circle would give when the telescope is pointed 
exactly to the pole,— technically, the polar point. The equator 
point is, of course, 90° from this. 

The nadir point is the reading of the circle when the tele- 
scope is pointed vertically downward. It is determined by the 
reading of the circle when the instrument is so set that the 
horizontal wire of the reticle coincides with its own image 
formed by a reflection from a basin of mercury placed on the 



pier below the instrument. To make this reflected image 
visible it is necessary to illuminate the reticle by light thrown 
towards the object-glass from behind the wires, — the ordinary The colii- 
illumination used during^ observation comes from the opposite "f^^^^^^s eye- 

... . piece. 

direction. This peculiar illumination is effected by what is 
known as the " collimating eye- 
piece." A thin glass plate inserted 
at an angle of 45° between the 
lenses of a Ramsden eyepiece 
throws down sufficient light, ad- 
mitted through a hole in the side 
of the eyepiece, and yet permits 
the observer to see the wires and 
their reflected image. The zenith 
point is, of course, just 180° from 
the nadir point thus determined. 

Obviously, the meridian-circle can 
be used simply as a transit, so that with 
this instrument and a clock the observer 
is in a position to determine both tJie 
right ascension and declination of any 
heavenly body that can be seen when 
it crosses the meridian. 


Fig. 25. — A 5-inch Altazimuth 
By Warner & Swasey 


70. Extra-Meridian Observa- 
tions. — Many objects, however, 
are not visible when they cross 
the meridian; a comet, for in- 
stance, or a planet, may be in 
such a part of the heavens that it transits only by daylight. To 
observe such objects we may employ a so-called universal instru- 
ment, or astronomical theodolite, which is simply an instru- The univer- 
ment with both horizontal and vertical circles like a large ^aimstru- 

^ ment, or 

surveyor's theodolite and is also called an altazimuth. By means altazimuth 
of this the altitude and azimuth of an object may be measured, 



tions with 
the equa- 

and, if the time is given, from these the right ascension and 
declination can be deduced. 

Fig. 25 shows the 5-iiich altazimuth of the Washington Observatory. 

More often, however, observations for the positions of bodies 
not on the meridian are made with the equatorial telescope 

already described, with which 
the difference between the right 
ascension and declination of 
the observed body and that 
of some star in its neighbor- 
hood is determined by means 
of a micrometer or, at present, 
often by photography. 

71. The Micrometer. — 
There are various forms of 
micrometers, the most common 
and generally useful being that 
known as the filar-position mi- 
crometer^ shown in Figs. 26 A 



c I I 

Fig. 26. — The Filar-Position Micrometer 

and B. It is a comparatively small instrument which is attached 
at the eye end of the telescope. It usually contains a set of fixed 
wires, two or three of them parallel to each other (only one, e, 
is shown in B^ which represents the internal construction 



of the instrument), crossed at right angles by a single line or 
set of lines. Under the plate which carries the fixed threads 
lies a fork moved by a carefully made screw with a graduated 
head, and this fork carries one or more wires parallel to the first 
set, so that the distance between the wires e and d (Fig. 26 ^) 
can be varied at pleasure and read off by means of the screw- 
head graduation. 

The box containing the wires is so arranged that it can itself 
be rotated around the op- 
tical axis of the telescope 
and set in any desired 
" position " ; for example, 
so that the movable wire 
d shall be parallel to the 
celestial equator when the 
position circle F should 
read 90°. When so set that 
the movable wire points 
from one star to another in 
the field of view, the '' po- 
sition angle " (see Fig. 191, 
Sec. 585) can be read off 
on the circle F. 

With such a micrometer 
we can measure at once the 
distance in seconds of arc 
between any two stars 

which are near enough to be distinctly seen in the same field of its use and 
view, and can determine the position angle of the line joining i^i^itations. 
them. The available range in a small telescope may reach 30'. 
In large telescopes, which with the same eyepieces give much 
higher magnifying powers, the range is correspondingly less, — 
not more than from 5' to 10'. When the distance between the 
objects exceeds 2' or 3', the filar micrometer becomes difficult 

Fig. 27. — Position Micrometer 
By Warner & Swasey 


to Use and inaccurate, because the observer cannot see both 
objects distinctly at the same time. 

Fig. 27 is a complete micrometer, fitted with electric illmnination. 

Theheiiom- 72. The Hellometer . — For the measurement of larger dis- 
eter:itscon- tanccs not exceeding two or three degrees the heliometer is used. 
This is a complete equatorially mounted telescope with its 
object-glass (usually from 4 to 8 inches aperture) diametrically 
divided into two halves which can be made to slide past each 
other for 3 or 4 inches (Fig. 28), the distance being measured 

on a delicate scale read by long microscopes 
which come down to the end of the instru- 
ment. The telescope tube can be rotated 
in its cradle so as to make the line of 
division of the lenses lie in any desired 

When the object-glass scale is at zero, 
the two half lenses act as a single lens 
and each object in the field of view pre- 

Method of 



'Si So 




Fig. 28. — The Heliometer 

sents a single image, as Sq and Mq in 

the figure » But as soon as one of the 
semi-lenses is pushed past the other, two images of each 
object appear, and the distance and direction between them 
can be varied at pleasure by sliding the lenses and rotating 
the tube. 

The distance between any two different objects is measured 
by making their images coincide (as, for instance, M-^ with /S'q, or 
A^2 with Jfo), and the observer does not have to "look two ways 
at once," nor is he obliged to trust to the stability of his instru- 
ment or the accuracy of the clockwork motion. 

On the whole, the heliometer stands at the head of astro- 
nomical instruments for the precision of its results and is 
employed in the most delicate investigations, like those upon 
solar and stellar parallax (Sees. 46T and 550). But it is a 


very complicated and costly instrument, and extremely laborious The rank 

, of the 

to use. , T 


The only one in the United States at present is the 7-inch among 
instrument at the Yale University Observatory. astronomi- 

cal instru- 

At present, however, such measurements oi the distance oi ments. 
an object from neighboring stars are very generally effected by 
means of jjliotography . Photographs of the field of view con- Observa- 
tainingf the obiect are made and afterwards measured, and in ^^^^^^y 

, , , . means of 

this case the limits of distance between the object and the stars photog- 
to which it is referred can be very much increased without ^^P^y- 
lessening the accuracy of the determination. 

73. The Sextant. — All the instruments so far mentioned, The sextant: 
except the chronometer, require some firmly fixed support, and *^® mstru- 
are therefore absolutely useless at sea. The sextant is the only mariner, 
one upon which the mariner can rely. By means of it he can 
measure the angular distance between two points (as, for 
instance, between the sun and visible horizon), not by pointing 
first to one and afterwards to the other, but by sighting them its peculiar 
both simultaneously and in apparent coincidence^ a " double advantage 

. . . over other 

image " measurement, in which respect the sextant is analo- instruments, 
gous to the heliometer. A skilful observer can make the 
measurement accurately even when he has no stable footing. 

Fig. 29 represents the instrument. Its -graduated limb is 
usually, as its name implies, about a sixth of a complete circle, 
with a radius of from 5 to 8 inches. It is graduated in its con- 
half degrees (which are, however, numbered as whole degrees) ^truction. 
and so can "measure any angle not much exceeding 120°. The 
index arm, or " alidade " (^MN in the figure), is pivoted at the 
center of the arc and carries a " vernier," which slides along 
the limb and can be fixed at any point by a clamp, with an 
attached tangent screw T. The reading of this vernier gives 
the angle measured by the instrument; the best instruments 
read to 10'' only, because it is impracticable to use a telescope 
with very much magnifying power. . 



Just over the center of the arc the index-mirror Jf, about 
2 inches by I2 in size, is fastened to the index arm, moving 
with it and keeping always perpendicular to the plane of the 
limb. At H the horizon-glass^ about an inch wide and about 
twice the height of the index-glass, is secured to the frame of 
the instrument in such a position that when the vernier reads 
zero the index-mirror and horizon-glass will be parallel to each 

Fig. 29. — The Sextant 

other. Only half of the horizon-glass is silvered, the upper half 
being left transparent. E is a small telescope screwed to the 
frame and directed towards the horizon-glass. 

If the vernier stands near, but not exactly at, zero, an observer 
looking into the telescope wdll see together in the field of view 
two separate images of the object towards which the telescope 
is directed; and if he slides the vernier, he will see that one of 
the images remains fixed while the other moves. The fixed 
image is formed by the rays which reach the object-glass directly 
through the unsilvered half of the horizon-glass ; the movable 


image, on the other hand, is produced by rays which have Double 
suffered two reflections, having* been reflected from the index- ^"^^se 

, , . formed by 

mirror to the horizon-glass and again reflected a second time sextant. 
from the lower, silvered half of the horizon-glass. When the 
two mirrors are parallel the two images coincide, provided the 
object is at a considerable distance. 

If the vernier does not stand at or near zero, an observer Angle 
looking- at an obiect directly throuofh the horizon-glass will see ^^^^®^'^ 

^ *' ^ . . two objects 

not only that object, but also, in the same telescopic field of whose 
view, whatever other obiect is so situated as to send its rays i^^^ges com- 

1 n ■ r 1- 17 7- ^^^^ equals 

to the telescope by reflection irom the mirrors ; and the reading half the 
of the vernier will give the angle at the instrument between the two ^^gi® 
objects whose images thus coincide^ — the angles between the niirrors. 
planes of the two mirrors being, as easily proved, just half the 
angle between the two objects, and the half degrees on the limb 
being numbered as whole ones. 

74. The principal use of the instrument is in measuring the altitude 
of the sun. At sea the observer usually proceeds as follows: first, setting Method of 
the index, loosely clamped, near zero and holding the sextant in his right observation 
hand with its plane vertical, he points the telescope towards the sun ; then ^^ ^^^' 
he slides the vernier along the arc with his left hand until he brings the 
reflected image of the sun down to the horizon, all the time keeping it in 
view in the telescope; finally, tightening the clamp and using the tangent 
screw, he makes the lower edge or limb of the sun jus't graze the horizon 
as he swings the sun's image back and forth by a slight motion of the instru- 
ment — it would be impossible on board ship to hold the image in contact 
with the horizon, and is not necessary. As soon as the contact is satis- 
factory he marks the time and afterwards reads the angle. The reading 
of the vernier after due corrections (see next chapter) gives the sun's true 
altitude at the moment. 

On land we have recourse to an " artificial horizon." This is a shallow Artificial 
basin of mercury covered with a roof of glass plates having their surfaces horizon used 
accurately plane and parallel. In this case we measure the angle between ^^ ' 
the sun and its image reflected in the mercury. The reading of the instru- 
ment corrected for index error then gives ticice the sun's apparent altitude. 

The skilful use of the sextant requires considerable dexterity, and from 
the low power of the telescope the angles measured are less precise than 



tion of the 
principle of 
the sextant. 

those determined by large fixed instruments, but the portability of the 
instrument and its applicability at sea render it invaluable. It was 
invented in practical form by Godfrey of Philadelphia, in 1730, though 
Newton, as was discovered by Halley, had really struck upon the same 
idea long before. 

75. The principle that the angle between the objects whose images 
coincide in the sextant is twice the angle between the mirrors (or between 
their normals) is easily demonstrated as follows : 

The ray SM (Fig. 30) coming from an object, after reflection first at M 
(the index-mirror) and then at H (the horizon-glass), is made to coincide 
with the ray OH coming from the horizon. 

From the law of reflection, we have the two angles SMP and PMH 

equal to each other, each being x. In the 
same way the two angles marked y are 
equal. From the geometric principle that 
the angle SMH, exterior to the triangle 
HME, is equal to the sum of the oppo- 
site interior angles at H and E, we get 
E = 2 X — 2 y. Similarly, from the tri- 
angle HMQ, Q = X — y ; whence E = 2 
Q = 2Q\ 

76. With the instruments above 
described all the fundamental obser- 
vations required in the investiga- 
tions of spherical and theoretical 
astronomy can be supplied, the sex- 
tant and chronometer being, however, the only ones available 
in nautical astronomy. 

Astrophysical studies require numerous physical instruments 
of an entirely different character, — spectroscopes, photometers, 
heat-measuring instruments, and various kinds of photographic 
apparatus. These will be considered later, as occasion arises. 

Fig. 30. — Principle of the Sextant 



1. If a firefly were to alight on the object-glass of a telescope, what 
would be the appearance to an observer looking through the instrument ? 
Would he think he saw a comet ? 

2. When a person is looking through a telescope, if you hold your 
finger in front of the object-glass, will he see it? 

3. If half the object-glass of a telescope pointed at the moon is covered, 
how will it affect the appearance of the moon as seen by the observer? 

4. If a certain eyepiece gives a magnifying power of 60 when used 
with a telescope of 5 feet focal length, what power will it give on a tele- 
scope of 30 feet focal length ? 

5. What is theoretically the angular distance between the centers of 
two star disks which are just barely separated by a telescope of 24 inches 
aperture (Sec. 46) ? 

6. Why is it important that the two pivots of a transit-instrument 
should be of exactly the same diameter? 

7. If the wires of a micrometer (Fig. 26) are so set that, used with a 
telescope of 10 feet focal length, a star moving along the right-ascension 
wire will occupy 15 seconds in passing from d to e, how long will it take- 
when the micrometer is transferred to a telescope of 50 feet focus ? 

8. If the threads of a micrometer screw are ^^ of an inch apart, what 
is the angular value of one revolution of the screw when the micrometer 
is attached to a telescope of 30 feet focal length ? 

9. Does changing the eyepiece of a telescope for the purpose of altering 
the magnifying power affect the value of the revolution of the microscope 
screw ? 



Dip of the 
depends on 
elevation of 
eye above 

Dip of the Horizon — Parallax — Semidiameter — Kefraction — Twinkling or Scintil- 
lation — Twilight 

Observations as actually made always require corrections 
before they can be used in deducing results. Those that 
depend on the errors or maladjustment of the instrument itself 
will not be considered here, but only such as are due to other 
causes external to the instrument and the observer. 

77. Dip of the Horizon In observations of the altitude of a 

heavenly body at sea, where the sextant measurement is made 

from the visible horizon, or sea-line^ it is 
necessary to take into account the depres- 
sion of the visible below the true astro- 
nomical horizon by a small angle called the 
dip. The amount of this dip depends upon 
the observer's altitude above the sea-level. 
In Fig. 31 C is the center of the earth, 
AB a portion of its level surface, and 
the eye of the observer at an elevation Ji 
above A. The line drawn perpendicular 
to OC is truly horizontal (regarding the 
earth as spherical), while the tangent OB 
is the line drawn from to B^ the visible horizon. The angle 
HOB is the dip, and is obviously equal to OCB. 
From the triangle OCB we have 

cos OCB= CB / CO = E / {B + h) = cos A, 

designating the dip by A. 


Fig. 31. — Dip of the 


The formula in this shape is inconvenient, because it deter- Formulae lor 
mines a small angle by means of its cosine. But since 1 — cos ^ A ^ ^® ^ ^^^" 
= 2 sin i A, we easily obtain the following : 

sin i A 


2{Fi, + h) 

Or, since A is always a small angle, and neglecting h in the 
denominator of the fraction as being insignificant compared 
with J?, we get 

sin A - 2 sin i A = 2 \/-^ = \-^- 

This gives with quite sufficient accuracy the true depression 
of the sea horizon as it would be if the line of sight were straight. 
But this is not the case, owing to refraction of the rays in pass- 
ing through the air, and the amount of this refraction is very 
uncertain and variable. Ordinarily the dip is diminished about 
one eighth of the amount computed by the formula. 

An approximate formula, obtained by substituting the radius 
of the earth (20 890000 feet) and reducing, gives A' (i.e., in 

2 h (feet) 

^^ ^^^\^^ , (Sec. 9), whence A' Approxi- 

20 890000 (feet) \ ^' J^te for- 

inula for 

= \h (feet) (nearly) ; or, in words, the dip in minutes of arc dip. 
equals the square root of the observer s elevation in feet; i.e., 
the dip is 1' at an elevation of 1 foot, 5' at an elevation of 
25 feet, 10' at an elevation of 100 feet, etc. 

This result is generally about five per cent too large, taking into account 
refraction ; but it is near enough for most practical purposes, since at sea 
the observer is seldom as much as 50 feet above the sea-level and cannot, 
with a sextant, measure altitudes more closely than to the nearest quarter 
of a minute. 

The formula A' = V 3 li (meters) agrees still more nearly with the actual 



The distance OB of the sea horizon is easily seen, from Fig. 31, to be 

rorinuia lor r. , ,. — — 

distance of i? tan A. An approximate formula is, distance in miles = ^'- ^^ ^• 

sea horizon. 

This, however, takes no account of refraction, and the actual distance is 
always greater. 

definition of 

Annual or 



Diurnal or 




78. Parallax (Fig. 32). — In general the word "parallax" 
means the difference between the direction of a heavenly body 
as seen by the observer and as seen from some standard point 
of reference. 

The annual or heliocentric parallax of a star is the difference 

of the star's direction as seen 
from the earth and from the sun. 
With this we have nothing to do 
for the present. 

The diurnal or geocentric paral- 
lax of the sun, moon, or a planet 
is the difference of its direction 
as seen from the center of the 
earth and from the observer's sta- 
tion on the earth's surface ; or, 
what comes to the same thing, 
it is the angle at the body made 
by two lines drawn from it, one to the observer, the other to 
the center of the earth. In Fig. 32 the parallax of the body 
F is the angle OPC, which equals xOF, and is the difference 
between ZOP and ZCF. Obviously this parallax is zero for a 
body directly overhead at Z, and a maximum for a body rising 
at H. Moreover, and this is to be specially noted, this paral- 
lax of a body at the horizon — the horizontal parallax — is 
simply the angular semidiameter of the earth as seen from the 
body. When we say that the moon's horizontal parallax is 57', 
it is equivalent to saying that, seen from the moon, the earth 
has an apparent diameter of 114'. 


' / 

/ ^ 


-i //P 



'/ / 






" \ 

Fig. 32. — Parallax 


79. Law of the Parallax. — From the triangle OCP we have 

PC : OC = sin COF : sin CFO, 
OT, B : r = sin f : sin p (since COF is the supplement of ^). 

This gives Formula 

sin ^ = - sin ?, (a) embodying 

^ R ^ the laws 

of diurnal 

or, since p is always a small angle, parallax. 

f = 206265'' - sin f. (^) 


When a body is at the horizon its zenith-distance is 90° and 
sin f=l. Hence, the horizontal parallax, 11, of the body is 
given by the formula 

n = — , (c) ; and p = U sin f. (d) 


Or, in words, the parallax at any altitude equals the horizontal 
parallax multiplied hy the sine of the apparent zenith-distance. 

From equation {c) we have also, for finding F^ the distance Relation 

of the body, • between dis- 

^ r ^ 206265 r ,, tanceofa 

F= . . or F = , (e) body and its 

Sm n n'' parallax. 

a relation of great importance as determining the distance of a 
heavenly body when its parallax is known. 

80. Equatorial Parallax. — Owing to the " ellipticity," or 
" oblateness," of the earth, the horizontal parallax of a body 
varies slightly at different places, being a maximum at the 
equator, where the distance of an observer from the earth's 
center is greatest. It is agreed to take as the standard the 
equatorial-horizontal-par allax^ i.e., the earth's equatorial semi- Equatorial 
diameter in seconds as seen from the body. parallax. 

If the earth were exactly spherical, the parallax would act 
in an exactly vertical plane and would simply diminish the 
altitude of the body without in the least affecting its azimuth. 



Effect of 
the earth's 
upon par- 
allax in the 
case of the 

' ' Augmen- 
tation ' ' of 
the moon's 

Formula for 
the aug- 

Really, however, it acts along great circles drawn from the geo- 
centrie zenith to the geocentric nadir (Sec. 11), and these circles 
are not identical with the vertical circles nor exactly normal to 
the horizon. For this reason the azimuth of the moon, which 
has a parallax of about a degree, is sensibly affected. The 
calculation of the parallax corrections to observations of the 
moon's right ascension and declination is also modified and 
greatly complicated. (See Campbell's Practical Astronomy, 
Sec. 26.) 

In the calculation of the parallax of all other bodies it is 
sufficient to regard the earth as spherical. 

81. Semidiameter. — In the case of the sun or moon the 
edge, or limh, of the object is usually observed, and to get the 
true position of its center the angular semidiameter must be 
added or subtracted. For all objects except the moon this may 
be taken directly from the ephemerides, but the moon's appar- 
ent diameter increases slightly with its altitude, being about 
eV P^^t' o^ about 30'^ greater when in the zenith than at the 
horizon, because at the zenith it is about 4000 miles, or gL- 
part of its whole distance from the center of the earth, nearer 
than at the horizon. At any observed zenith-distance, OP 
(Fig. 32), the apparent or "augmented" semidiameter (s'), as 
seen from 0, is greater than the semidiameter (s) given in the 
ephemeris as seen from C, in the ratio of PC to PO. From 
the triangle POC we obtain, therefore, 

s' :S'.:PC:PO:: sin POC : sin PCO 
(f being the apparent zenith-distance). 

sin f 

sin f : sin {^— p) 


s' = s 

sin {^ — p) 

This "augmentation" of the moon's diameter, amounting to about 
30'' near the zenith, has, of course, nothing whatever to do with the opti- 
cal illusion already referred to which makes the moon seem larger when 
near the horizon. 


82. Refraction. As the rays of light from a star enter our 

atmosphere, unless they strike perpendicularly they are bent 
downwards by refraction and follow a curved path, as illus- 
trated in Fig. 33. 

Since the object is seen in the direction from which the rays 
enter the eye, the effect is to make the apparefit altitude of the 
object greater than the true. 

Refraction, like parallax, is zero at the zenith and a maxi- 
mum at the horizon, where under average conditions it lifts an 
object about 35', leaving the azimuth, however, unchanged. But 
the law of refraction is very different 
from that of parallax. 

Its amount depends upon the den- 
sity of the air (which is determined 
by the barometric pressure and tem- 
perature) as well as the altitude of 
the object, but is independent of its 

The theory of refraction is too 
complicated to be discussed here, and 

the reader is referred to Campbell's or Chauvenet's Practical 

The computation of the correction when precision is required 
is made by means of elaborate tables provided for the purpose and 
given in works on practical astronomy, the data being the observed 
altitude of the object, the temperature, and the height of the 
barometer. Increase of atmospheric pressure slightly increases 
the refraction, and increase of temperature diminishes it. 

For altitudes exceeding 25° the following approximate for- 
mula, corresponding to a temperature of zero Centigrade 
(32° Fahrenheit) and a barometric pressure of 30 inches, may 
be used, and will generally give results correct within a few 
seconds, viz., r" = 60". 7 tan f, in which f is the apparent zenith- 

Fig. 33. — Atmospheric Refraction 


Its effect to 
increase the 
altitude of a 

Affected by 
and baro- 

mate for- 
mulae for 
bodies above 
15° altitude. 



table in 

Effect of 
refraction to 
length of 
the day at 
expense of 
the night. 

The following formula (due to Professor Comstock) is a little 
more complicated, but much more accurate, viz.. 



983 5 
460 + ^ 

tan f, 

in which h is the height of the barometer in inches and t is the 
temperature on Fahrenheit^ § scale. For altitudes above 15° 
this formula will seldom be over V in error. 

The little Table VIII (Appendix) gives by inspection pretty 
accurately the refraction under the circumstances stated in its 
heading ; and by applying the approximate corrections for 
barometer and thermometer indicated in the note below it, the 
results will seldom be more than 2" in error. 

It is hardly necessary to add that this refraction correction, 
required by most astronomical observations of position, is very 
troublesome, and. usually involves more or less uncertainty 
and error from the continually changing and unknown condi- 
tion of the atmosphere along the path followed by the rays of 

For methods by which the amount of the refraction is deter- 
mined hy observation., the reader is referred to works on practical 
astronomy, or to the author's General Astronomy^ Art. 94. 

83. Effect of Refraction near the Horizon. — The horizontal 
refraction, ranging as it does from 32' to 40', according to 
meteorological conditions, is always somewhat greater than the 
diameter of either the sun or the moon. At the moment, 
therefore, when the sun's lower limb appears to be just rising 
or setting, the whole disk is really below the plane of the hori- 
zon; and the time of sunrise in our latitudes is thus accelerated 
from two to four minutes, according to the inclination of the 
sun's diurnal circle to the horizon, which varies with the time 
of the year. Of course, sunset is delayed by the same amount, 
and thus at both ends the day is lengthened at the expense of 
the night. 


Near the horizon the refraction changes very rapidly ; while Effect of 
under ordinary summer temperature it is about 35' at the hori- refraction 

• • nn/ • J" 1 li- 1 upon the 

zon, it is only 29' at an elevation oi halt a degree, so that as form of the 
the sun or moon rises the bottom of the disk is lifted 6' more '^^^^^ ^^ "^"^^ 
than the top and the vertical diameter is thus made apparently ^j^g^ yg^.„ 
about one-fifth part shorter than the horizontal. This quite "ear the 
notably distorts the disk into the form of an oval flattened 
on the under side. In cold weather the effect is much more 

Two other semi-astronomical effects, the twinkling of the 
stars and twilight, are due to the action of our atmosphere, and 
may be treated in this connection, though in no other way con- 
nected with the principal subject of the chapter. 

84. Twinkling or Scintillation of the Stars. — - This is a purely Scintiiia- 
atmospheric phenomenon, usually conspicuous near the horizon, ^^^^^ ^^^ 

. "^ atmospheric 

where it is often accompanied by marked changes of color, phenome- 
Near the zenith it generally disappears, and at other altitudes ^^'^• 
it differs greatly on different nights. As a rule only the sfars 
twinkle strongly ; the planets, Mercury excepted, usually shine 
with an almost steady light. 

Authorities differ as to the details of explanation, but prob- 
ably scintillation is mainly due to two cooperating causes, both 
depending on the fact that the air is generally full of streaks 
and wavelets of unequal density carried by the wind. 

(1) Light coming through such a medium is concentrated in Unequal 
some places and diverted from others by simple refraction, like ^'ef^'^^^ions 

. . . by drifting 

light from an electric lamp shining through an ordinary window- atmospheric 
pane upon the opposite wall. If the light of a star were strong wavelets of 
enough, a white surface illuminated by it would be covered by density. 
bright and dark mottlings, drifting with the wind ; and as such 
mottlings pass the eye the star appears to fade and brighten by 
turns. Looked at in the telescope, it also "dances," being 
slightly displaced back and forth by the irregular refraction. 



tary action 
of optical 
' ' inter- 

Effect upon 
the spec- 
trum of a 

Why planets 
do not 

Cause of 

Duration of 

(2) The other cause of twinkling is optical interference. Pen- 
cils of light coming from a star (optically a mere luminous 
point) reach the observer's eye by routes differing only slightly, 
and are just in a condition to " interfere." The result is 
the temporary destruction of rays of certain wave-lengths and 
the reinforcement of others. Accordingly, the " spectrum " 
(Sec. 569) of a twinkling star is traversed by dark bands in the 
different colors, oscillating back and forth, but, on the whole, 
when the star is rising, progressing from the blue towards the 
red, and vice versa when the star is near the setting. 

The planets do not twinkle, because they are not luminous 
points^ but have disks made up of a congeries of such points ; 
while each point twinkles like a star, the twinklings do not 
synchronize with each other, and so the general sum of light 
remains practically uniform. When very near the horizon, how- 
ever, the irregular refraction is sometimes sufficiently violent 
to make them dance and change color. Since the disk of 
Mercury is very small, and the planet is never seen except 
near the horizon, it usually behaves like a star. 

85. Twilight. — This is caused by the reflection of sunlight 
from the upper portion of the earth's atmosphere, perhaps from 
the air itself, perhaps from the minute solid particles in the air, 
— authorities differ. After the sun has set, its rays, passing 
over the observer's head, still continue to shine through the air 
above him, and twilight continues as long as any portion of the 
illuminated air remains in sight from where he stands. It is con- 
sidered to end when stars of the sixth magnitude become visible 
near the zenith, which does not occur until the sun is about 
18° below the horizon; but this varies considerably for different 
places, according to the purity of the air. 

The length of time required by the sun after setting to reach this depth 
varies with the season and with the observer's latitude. In latitude 40° it 
is about ninety minutes on March 1 and October 12, but more than two 
hours at the summer solstice. In latitudes above 50°, when the days are 


longest, twilight never disappears even at midnight. On the mountains 
of Peru, on the other hand, it is said never to last more than half an hour, 
probably because the upper air in that region is practically clear from dust 

From the fact that twilight lasts until the sun is 18° below the horizon. Height of 
the height of the twilight-producing atmosphere can easily be computed, ^^^ earth's 
and comes out about 50 miles. This, however, is not the real limit of ^ ii^ospiere. 
the atmosphere. The phenomena of meteors show that at an elevation 
of 100 miles there is still air enough to resist their motion and cause their 

Soon after the sun has set, the twiligld how appears rising in the east, — The twilight 
a dark blue segment, bounded by a faintly reddish arc. It is the shadow ^ow. 
of the earth upon the air, and as it rises the arc becomes rapidly diffuse 
and indistinct and is lost long before it reaches the zenith. 


1. What is the approximate dip of the horizon from a hill 900 feet 
high (Sec. 77) ? 

2. How high must a mountain be in order that the dip of the horizon 
from its summit may be 2°? 

3. What is the distance of the horizon in miles, as seen from the 
summit of this mountain (Sec. 77)? 

4. Assuming the horizontal parallax of the sun at 8'^8, what is the 
horizontal parallax of Mars when nearest us, at a distance of 0.378 astro- 
nomical units? (The astronomical unit is the distance from the earth to 
the sun.) 

5. What is the greatest apparent diameter of the earth as seen from 

6. What is the horizontal parallax of Jupiter when at a distance of 6 
astronomical units? 

7. Does atmospheric refraction increase or decrease the apparent size 
of the sun's disk when it is near the horizon? 

8. What is the lowest latitude where twilight can last all night? Can 
it do so at New York? at London? at Edinburgh? 



Latitude — Time — Longitude • 

Azimuth — The Right Ascension and Declination of 
a Heavenly Body 

86. There are certain problems of practical astronomy which 
are encountered at the very threshold of all investigations 

problems of 

observation, respecting the heavenly bodies, the earth included. The student 
must know how to determine his positio7i on the surface of the 
earthy that is, his latitude and longitude ; how to ascertain the 
exact time at which mi observation is made; and how to observe 
the precise position of a heavenly/ body and fix its right ascen- 
sion and declination. 

87. Definitions of the Observer's Latitude. — In geography 
the latitude of a place is usually defined simply as its distance 
north or south of the equator, measured in degrees. This is 
not explicit enough unless it is stated how the degrees them- 
selves are to be measured. If the earth were a perfect sphere 
there would be no difficulty, but since the earth is sensibly 
flattened at its poles the geographical degrees have somewhat 
different lengths in different parts of the earth. The funda- 
mental definition of astronomical latitude has already been 
given (Sec. 32) as the angle between the direction of gravity 
where the observer stands and the plane of the equator. The 
angle between gravity and the eartKs axis is the colatitude of 
the place. Other equivalent definitions of the latitude are the 
altitude of the pole and the declination of the zenith., which is 
the same as the altitude of the pole, as is clear from Fig. 34, 
where ZQ obviously equals NP. 


of astronom 
ical latitude 



The problem, then, is to determine by observation of the heav- 
enly bodies either the angle of elevation of the celestial pole^ or the 
distance in degrees hetiveen the zenith and the celestial equator. 

88. First Method : by Observation of Circumpolar Stars. — 
The most obvious method (already referred to) is by observing 
with a suitable instrument the altitude of some star near the 
pole at the moment when it is crossing the meridian above 
the pole, and again twelve sidereal hours later when it is once 
more on the meridian but below the pole. In the first case its 
altitude is the greatest possible ; in the second, the least. The 
mean of the tivo altitudes (each corrected for atmospheric refrac- 
tion) is the altitude of the pole or the latitude of the observer. 

The method has the great advantage that it is an independe7it 
one ; that is, the observer is not obliged to depend upon his 
predecessors for any of his data. But the method fails for 
stations very near the 

equator, because there - «- — -^ '« 

the j)ole is so near the 
horizon that the neces- 
sary observations cannot 
be made. 

At an observatory the 
observations are usually 
made with the meridian- 
circle, and the mean of a 

great number of observations is necessary in order to elimi- 
nate the slight errors in the computed refraction corrections 
due to varying atmospheric conditions. Where the meridian- 
circle is not available, the observations may also be made 
with a sextant or theodolite, but the results are much less 

89. Second Method : by the Meridian Altitude or Zenith- 

Distance of a Body whose Declination is accurately known 

In Fig. 34 the circle JIQPX is the meridian, Q and F being 

Latitude by 
of circum- 
polar stars. 

and disad- 
vantages of 
the method. 

Fig. 34 



Latitude by 
altitude of 
object of 

Formula for 
latitude in 
this case. 

and disad- 
A-antages of 
this method. 

Latitude at 

sea by 
of the sun. 

respectively the equator and the pole and Z the zenith. QZ is 
the declination of the zenith, or the latitude of the observer. If, 
when the star s crosses the meridian, we observe its zenith- 
distance, ^3 {Z8 in the figure), its declination, Qs or 8^ being 
known, then evidently ()Z equals Q% plus %Z\ that is, the latitude 
equals the declination of the star plus its zenith-distayice. If the 
star were at s\ south of the equator, the same equation would 
still hold algebraically^ because the declination Qs^ is then a 
negative quantity ; and if the star were at n between the zenith 
and pole, we should have its north zenith-distance, ?'^, a negative 
quantity. In all cases, therefore, we may write </> = S + f . 

If we use the meridian-circle in making our observations, we can 
always select stars that pass near the zenith, where the refraction is small, 
which is in itself a great advantage. Moreover, we can select them in 
such a way that some will be as much north of the zenith as others are 
south, and this will practically eliminate even the slight refraction errors that 
remain. On the other hand, in using this method we have to obtain our 
star declinations from the catalogues made by previous observers, so that 
the method is not an " independent " one. 

90. At sea the latitude is usually obtained by observing with 
the sextant the sun^s maximum altitude, which occurs, of course, 
at noon. Since at sea one seldom knows beforehand precisely 
the moment of local noon, the observer takes care to begin his 
observations some minutes earlier, repeating his measure of the 
sun's altitude every minute or two. At first the altitude will 
keep increasing, but immediately after noon occurs it will begin to 
decrease. The observer uses, therefore, the maximum ^ altitude 
obtained, which, corrected for refraction, parallax, semidiameter, 
and dip of the horizon, will give him the true meridian altitude 
of the sun. The Nautical Almanac gives him its declination.. 

iQn account of the sun's motion in declination and the northward or 
southward motion of the ship itself, the sun's maximum altitude is usually 
attained not precisely on the meridian, but a short time earlier or later. This 
requires a slight correction to the deduced latitude, the calculation of which 
is explained in books on navigation. 


91. Third Method: by Circummeridian Altitudes. — If the Latitude by 
observer knows his time with reasonable accuracy, he can obtain ^■^^■^"'^- 


his latitude from observations of the altitude of a heavenly body altitudes. 
made when it is near the meridian with practically the same 
precision as at the moment of meridian passage. It lies beyond 
our scope to discuss the method of reduction, which is explained, 
with the necessary tables, in all works on practical astronomy. 

The great advantage of the method is that the observer is 
not restricted to a single observation at each meridian passage 
of the sun or of the selected star, but can utilize the half -hours 
preceding and following that moment. The meridian-circle, of 
course, cannot be used. Usually the sextant, or a so-called 
"universal instrument" (Sec. TO), is employed. 

92. Fourth Method^ : by the Zenith-Telescope. — The essential Latitude by 
characteristic of the method is the measurement Avith a microm- *^^f zemth- 

telescope — 
eter of the d iffe r e n e e hettveen the nearly equal zenitli-distances the most 

of two stars which pass the meridian within a few minutes accurate 
of each other, one north and the other south of the zenith, and 
not very far from it; such pairs of stars can now always be 
found in our star-catalogues. 

A special instrument, known as the zenith-telescope^ is gen- 
erally employed, though a simple transit-instrument, provided 
with reversing apparatus, a delicate level attached to the tele- 
scope, and a declination micrometer is now often used. 

Fig. 3.5 shows the very complete zenith-telescope of the Flower Observa- 
tory near Philadelphia. 

At the GeorgetoT\Ti Observatory a photographic zenith-telescope is used, 
ha^dng a photographic plate in place of the eyepiece. 

The telescope is set at the proper altitude for the star which Method of 
first comes to the meridian and the "latitude level," as it is 
called, — which is attached to the telescope — is set horizontal ; 

1 Known as the "American method." because first practically introduced 
by Captain Talcott, of the United States Engineers, in a boundary survey in 
1845. It is now very generally adopted and considered the best. 



in dispens- 
ing with a 

as the star passes through the field of view its distance north 
or south of the central horizontal wire is measured by the 
micrometer. The instrument is then reversed so that the tele- 
scope points towards 
the north (if it was 
south before), and 
the telescope so 
readjusted, if neces- 
sary, that the level 
is again horizontal, 
— taking great care, 
however, not to 
disturb the angle 
between the level and 
the telescope itself. 
The telescope is then 
evidently elevated 
at exactly the same 
angle as before, but 
on the opposite side 
of the zenith. As 
the second star 
passes through the 
field, we measure 
with the micrometer 
its distance north or 
south of the central 
wire. The compari- 
son of the two measures gives the difference of the two zenith- 
distances with great accuracy and without the necessity of 
depending upon any graduated circle. 

In field operations like those of geodesy this is an enormous 
advantage, both as regards the portability of the instrument 
and the attainable precision of results. . - 

Fig. 35. — a Zenith-Telescope 
By Wai'uer tfe Swasey 


From Fig. 34 we have 

for star south of zenith, <^ = 8^ + f j 
for star north of zenith, (/> = 8,^ — f,^. 

Adding the two equations and dividing by 2, we have 

i^/^.s + ^«\ , /C~C\ Formula lor 

9 / V 9 / the latitude. 

The star-catalogue gives us the declinations of the two stars 
(S.H-8^J ; and the difference of the zenith-distances (51. — f,J is 
determined by the micrometer measures. 

When the method was first introduced it was difficult to find 
pairs of stars whose declination was known with sufficient pre- 
cision. At present our star-catalogues are so extensive and 
exact that this difficulty has practically disappeared. 

Refraction is almost eliminated, because the two stars of each Refraction 
pair are at very nearly the same zenith-distance. eliminated. 

Evidently the accuracy depends ultimately upon the exactness with 
which the level measures the slight but inevitable difference between the 
inclinations of the instrument when pointed on the two stars. 

In Dr. Chandler's Almucantar (Sec. 67) the telescope preserves its 
constant declination automatically ^ by being mount-ed upon a base which 
floats in mercury, thus dispensing with the level. 

There are numerous other methods for obtaining the latitude. In 
Chauvenet's Practical Astronomy over forty are given, some of which can 
fairly compete in precision with those named above. 

93. The Gnomon. — The ancients could not use any of the Ancient 
preceding- methods for finding the latitude. They were, how- ^^^^^^^^ of 

^ <=> o J ' determining 

ever, able to make a very respectable approximation by means the latitude 
of the simplest of all astronomical instruments, the gnomon. ^J^^^ 
This is merely a vertical shaft or column of known height 
erected on a perfectly horizontal plane, and the observation 



consists in noting the length of the shadow cast at noon 
at certain times of the year. Suppose, for instance, that on 
the day of the sumone?- solstice, at noon, the length of the 
shadow is AC (Fig. 36). The height AB being given, we can 
easily compute in the right-angled triangle the angle ABC, 
which equals SBZ, the sun's zenith-distance when farthest 

Again, observe the length AD of the shadow at noon of the 
shortest day in winter and compute the angle ABD, which is the 
sun's corresponding zenith-distance when farthest south. Now, 
since the sun travels equal distances north and south of the 

celestial equator, the mean 
of the two zenith-distances 
will give the angular dis- 
tance between the equator 
and the zenith, i.e., the 
declination of the zenith, 
which is the latitude of 
the place. 

The method is an inde- 
pendent one, like that of 
the observation of circum- 
polar stars, requiring no 
data except those which 
the observer determines for himself. It does not admit of much 
accuracy, however, since the penumbra at the end of the shadow 
makes it impossible to measure its length very precisely. 

It should be noted that the ancients, instead of designating the position 
Clhnate." of a place by means of its latitude, used its climate; the climate (from 
KXifxa) being the slope of the plane of the celestial equator, the angle 
AEB, which is the colatitude. 

For the use of the gnomon in determining the obliquity of the ecliptic 
and the length of the year, see Sees. 164 (2) and 182. Many of the Egyp- 
tian obelisks are known to have been used for astronomical observations, 
and perhaps were erected mainly for that purpose. 

C -E 

Fig. 36. — Latitude by the Gnomon 



+ 0.20 



■0.20 -0.30 , 



94. Variation of Latitude and Motion of the Poles of the Earth. 
— It has long been doubted whether latitudes are strictly con- 
stant. They cannot be so if the axis of the earth shifts its 
position within the globe. Some have supposed that in the 
past there have been great changes of this kind, seeking thus 
to explain certain geological epochs, as, for instance, the glacial 
and the carboniferous. But thus far no evidence of any consid- 
erable displacement has appeared, nor is there any satisfactory 
proof of certain slow, 
continuous " secular " 
changes, which have 
been strongly sus- 

Theoretically, how- 
ever, any alteration in 
the arrangement of the 
matter of the earth, 
by elevation, subsi- 
dence, transportation, 
or denudation, must 
necessarily disturb the 
axis and change the """"" 
latitudes to some ex- 
tent. The question 
is merely whether our observations can be made sufficiently 
accurate to detect the change. Since 1889 the limit has been 
reached, and we now have conclusive proof of such effects. 

The first satisfactory evidence of the fact was obtained at 
Berlin by Kiistner, and at other German stations in 1888 and 
1889, and the result has since been abundantly confirmed by 
observations at many other stations. Moreover, Dr. S. C. 
Chandler of Cambridge, U.S., by a brilliant and laborious series 
of investigations, finds the same variations clearly exhibited in 
almost every extended body of reliable observations made since 

+ 0.10 



+ 0.10 


+ 0.20 

+ 0.10 


Fig. 37 





No evidence 
of any 
changes in 
the position 
of the 
earth's axis. 

must occur. 

First obser- 
in 1888. 



Nature of 
the periodic 
motion of 
the pole. 

1750. From the whole mass of evidence he concludes that 
the movement of the pole at present is composed of two 
motions, — one an annual revolution in an ellipse about 30 
feet long, but varying in width and position, the other a revo- 
lution in a circle about 26 feet in diameter and having a period 
of about 4-^8 days, — both revolutions being counter-clockwise. 
The resultant motion presents a very irregular appearance and 
changes greatly from year to year. 

Fig. 37 represents the actual motion from 1890 to 1898 as deduced by 
Albrecht from all available observations. 

The annual component of this polar motion is very likely due to meteoro- 
logical causes which follow the seasons, such as the deposit of rain, snow, 
and ice. The explanation of the 428-day component is not yet entirely 
clear, and its discussion would take us too far. 

It is likely also that irregular disturbances, due to various causes — for 
instance, perhaps, earthquakes — may modify the regular periodic motions. 

Time de- 
fined as 

tion of time. 

The three 
kinds of 

95. Different Kinds of Time. — Time is usually defined as 
measured duration. From the beginning the apparent diurnal 
rotation of the heavens has been accepted as the standard unit, 
and to it we refer all artificial measures of time, such as clocks 
and watches. 

In practice the accurate determination of time consists in 
finding the Hour Angle (Sec. 21) of the object or point which 
has been selected to mark the beginning of the day by its jjassage 
across the meridian. 

In astronomy three kinds of time are now recognized : side- 
real time., apparent solar time, and mean solar time., — the last 
being the time of civil life and ordinary business, while the 
first is used for astronomical purposes exclusively. Apparent 
solar time (formerly called true time) has now practically fallen 
out of use, except in countries where watches and clocks are 
scarce or unknown and sun-dials are the ordinary timekeepers. 


96. Sidereal Time. — The celestial object which determines Sidereal 
sidereal time by its position in the sky at any moment is, it ^"^i« — ^i^e 

^ ^ . . ho^^i" angle 

will be remembered, the vernal equinox oy first of Aries (symbol, of the vernal 
0(°), ^.e., the point where the sun crosses the celestial equator equinox. 
in the spring, about March 21 every year. 

As already stated (Sec. 25), the local sidereal day'^ begins at 
the moment when the first of Aries crosses the observer's 
meridian, and the sidereal time at any moment is the hour 
angle of the vernal equinox; i.e., it is the time marked by a 
clock so set and adjusted as to show sidereal noon (0^^0™0^) at 
each transit of the first of Aries. 

The equinoctial point is, of course, invisible ; but its posi- 
tion among the stars is always known, so that its hour angle at 
any moment can be determined by observing the stars. 

97. Apparent Solar Time. — Just as sidereal time is the hour Apparent 
angle of the vernal equinox, so apparent solar time at any solar time — 
moment is the hour angle of the sun. It is the time shown hy angle of the 
the su7i-dial, and its noon occurs at the moment when the sun's s^^' if^e"ti- 

cal with sniJ- 

center crosses the meridian. ^ji^i time. 

On account of the earth's orbital motion (explained more 
fully in Chapter VI), the sun appears to move eastward along 
the ecliptic, completing its circuit in a year. Each noon, ^^ 

therefore, it occupies a place among the stars about a degree 
farther east than it did the noon before, and so comes to the 
meridian about four minutes later, if time is reckoned by a 
sidereal clock. In other words, the solar day is about four 
minutes longer than the sidereal, the difference amounting to 
exactly one day each year, which contains 366 i sidereal days. 

But the sun's eastward motion is not uniform, for several 

1 On account of the precession of the equinoxes (to be discussed later) , the 
sidereal day thus defined is slightly shorter than it would be if defined as the 
interval between successive transits of some given star ; the difference being a 
little less than yi-g- of a second, or one day in 25800 years, — too little to be 
worth taking into account in any ordinary calculation. 



solar clays 
vary in 
Hence ap- 
parent time 
is unsatisfac- 

The ficti- 
tious sun. 

Mean solar 
time : the 
hour angle 
of the ficti- 
tious sun. 

time unsuit- 
able for 

with appar- 
ent solar 

of mean 
solar time. 

reasons, and the apparent solar days therefore vary in length. 
December 23, for mstance, is about fifty-one seconds longer 
from sun-dial noon to noon again (by a sidereal clock) than Sep- 
tember 16. For this reason apparent solar or sun-dial time is 
unsatisfactory for scientific use and cannot be kept by any simple 
mechanical arrangement in clocks and watches. At present it 
is practically discarded in favor of mean solar time. 

98. Mean Solar Time — A fictitious sun is, therefore, imagined, 
moving uniformly eastioard in the celestial equator and complet- 
ing its annual course in exactly the same time as that in which 
the actual sun makes the circuit of the ecliptic. This fictitious 
sun is made the timekeeper for mean solar time. It is mean 
7ioon when its center crosses the meridian, and at any moment 
the hour angle of the fictitious sun is 'the mean time for that 
moment. The mean solar days are, therefore, all of exactly 
the same length and equal to the length of the average apparent 
solar day, the mean solar day being longer than the sidereal by 
3^^55^.91 (mean solar minutes and seconds) and the sidereal day 
shorter than the solar by 3"'56^.55 (sidereal minutes and seconds). 

99. Sidereal time will not answer for business purposes, 
because its noon (the transit of the vernal equinox) occurs 
at all hours of the day and night in different seasons of the 
year : on September 22, for instance, it comes at midnight. 
Apparent solar time is unsatisfactory because of the variation 
in the length of its days and hours. Yet we have to live 
by the sun: its rising and setting, daylight and night, control 
our actions. 

Mean solar time furnishes a satisfactory compromise. It has 
a time unit which is invariable, and it can be kept by clocks and 
watches, while it agrees nearly enough with sun-dial time for 
convenience. It is the time now used for all purposes except 
in some kinds of astronomical work. 

The difference between apparent time and mean time (never 
amounting to more than about a quarter of an hour) is called 


the equation of time and will be discussed hereafter in connec- Equation of 

tion with the earth's orbital motion (Sec. 174). *™®' 

Since there are 365.2421 solar days in a year (Sec. 182) and Relation 

one more sidereal day, we have the following^ fundamental rela- ^®^^^®®^^ ^]^® 

*^ ° number of 

tion: — the number of sidereal seconds in any time interval : the sidereal and 

number of mean solar seconds in the same interval : : 366.24^1 : "^®^^ ^^^^^ 

seconds in a 

Oo5.24'21. given time 

From this it follows at once that to reduce a solar time interval. 
interval to sidereal, we must divide the number of seconds it Reduction of 
contains by 365.2421, and add the quotient to the number of ^^1^^^*"^^^ 

^ ^ interval to 

solar seconds. To reduce a sidereal interval to solar, divide by sidereal, and 
366.2421, and subtract the quotient from the number of sidereal ^^^^ versa. 

The Nautical Almanac gives the sidereal time of mean solar 
noon for every day of the year, with tables by means of which 
mean solar time can be accurately deduced from the correspond- 
ing sidereal time, or vice versa, by a very briefs calculation. 

100. The Civil Day and the Astronomical Day. — The astro- Theastro- 
nomical day begins at mean noon; the civil day, twelve hours ^^^^^^^^ ^"^^ 

, . . . . civil days. 

earlier at midnight. Astronomical mean time is reckoned 
around through the whole twenty-four hours instead of being 
counted in two series of twelve hours each : thus, 10 A.M. of 
Wednesday, February 27, civil reckoning, is Tuesday, February 
26, 22 o'clock, by astronomical reckoning. This must be borne 
in mind in using the Almanac.^ 

1 The approximate relation betw^een sidereal time and mean solar time is very- 
simple. Assuming that on March 22 the two times agree, after that, day the 
sidereal time gains tioo hours each month. On April 22, therefore, the sidereal 
clock is two hours in advance, on June 22, six hours in advance, and so on. 
On account of the differing length of months, this reckoning is slightly errone- 
ous in some parts of the year, but is usually correct within four or five minutes. 
March S2 is taken as the starting-point because it distributes the errors better 
than the 21st. For the odd days the gain may be taken as four minutes daily. 

2 The astronomical day is made to begin at noon because astronomers are 
"night-birds," and would find it inconvenient to have to change dates at 
midnight in the middle of their work. 



tion of time 
consists in 
the error 
of a time- 

tion of time 
by tlie 


to observe a 
number of 
stars in 
order to 
attain higli 


In practice the problem of determining time always consists 
in ascertaining the error or correction of a timepiece, i.e., the 
amount by which the clock or chronometer is faster or slower 
than the time it ought to indicate. 

101. Determination of Time by the Transit-Instrument. — The 
method most employed by astronomers is by observations with 
the transit-instrument (Sec. 61). We observe the time shown by 
the sidereal clock at which a star of known right ascension crosses 
each wire of the reticle. The mean is taken as the instant of 
crossing the instrumental meridian, and when the instrument 
is in perfect adjustment the difference between the star's right 
ascension and the observed clock time will be the clock's 
"error"; or, as a formula, At = a-t, — At being the usual 
symbol for the clock error, and t the observed time. 
^ The Almanac supplies a list of several hundred stars whose 
right ascension and declination are accurately given for every 
tenth day of the year, so that the observer at night has no diffi- 
culty in finding a suitable star at almost any time. In the day- 
time he is, of course, limited to the brighter stars. 

The observation of a single star with an instrument in ordi- 
nary adjustment will usually give the error of the clock within 
half a second; but it is much better and usual to observe a 
number of stars, reversing the instrument upon its Y's once at 
least during the operation. This will enable him to determine 
and allow for the faults of instrumental adjustment, so that 
with a good instrument a skilled observer can thus determine 
his clock error within about a thirtieth of a second of time, 
provided proper correction is applied for his " personal equa- 
tion" (Sec. 64). 

If instead of observing a star we observe the sun with this 
instrument, the time as shown by the mean solar clock ought to 
be twelve hours plus or minus the equation of time as given ni 


the Almanac. But for various reasons transit observations of Soiar time 
the sun are less accurate than those of the stars, and it is far i]«"^^^y "^>w 


better to deduce the mean solar time from the sidereal time by from 
means of the almanac data. sidereal. 

102. The Method of Equal Altitudes. — If we observe the time 
shown by the chronometer or the clock when a star attains a Method of 
certain altitude and then the time when it attains the same ^*,V!f\ 


altitude on the other side of the meridian, the mean of the two 
times will be the time of the star's transit across the meridian, 
provided, of course, that the chronometer runs uniformly during 
the interval. 

We may also use stars of slightly differing declination, one Modification 
on one side of the meridian, and the other observed a few ^^^^^^ 

method in 

minutes later on the other side; and by a somewhat tedious observing 

calculation it is possible to determine the error of the clock ^^^^^ ^^ 

with practically the same accuracy as if both observations had different 

been made on the same star, and much more quickly. declination. 

If we observe the sun in this manner in the morning, and 

again in the afternoon, the moment of apparent noon will seldom Correction 

be exactly half-way between the two observed times, and proper ^^^i^^^'^d in 

'' 'J -> L L deducing- 

correction must be made for the sun's slight motion in decli- time from 

nation during the interval, — a correction easily computed by ^^^^^^ ^^^^" 

tudes of the 

tables lurmshed lor the purpose. • sun. 

The advantage of this method is that the errors of gradua- 
tion of the instrument have no effect, nor is it necessary for the 
observer to know his latitude except approximately. 

On the other hand, there is, of course, danger that the second 
observation may be interfered with by clouds. Moreover, both 
observations must be made at the same place. 

103. Marine Method: by a Single Altitude of the Sun, the Observation 
Observer»s Latitude being known. — Since neither of the preced- todetermhie 

'^ ^ time at sea. 

ing methods can be used at sea, the following is the method 
usually practised. The altitude of the sun, at some time when 
it is rapidly rising or falliyig (i.e., not near noon), is measured 



tion of the 
time from 
the obser- 

with the sextant, and the correspondmg tune shown by the 
chronometer accurately noted. 

We then compute the hour angle of the sun, P, from the 
triangle PZS (Fig. 38), and this hour angle, corrected for the 
equation of time, gives the mean solar time at the observed 
moment. The difference between this time and that shown 
by the chronometer is the error of the chronometer on local 


In the triangle ZPS (which is the same as SFO in Fig. 8) all 

three of the sides are given: PZ is the complement of the 

latitude <^, which is sup- 
posed to be known; JP*S^ 
is the complement of the 
sun's declination S, which 
is found in the Alma7iac, as 
is also the equation of time; 
while ZS or f is given' 
by observation, being the 
complement of the sun's 
altitude as measured by the 
sextant and corrected for dip, semidiameter, refraction, and 
parallax. The formula ordinarily used is 

Time when 
should be 



Fig. 38. — Determination of Time by the 
Sun's Altitude. 


_ / sin ir? + (c^ - g)]sm \\i-{4> - g)] 
^ cos (/) cos h 

In order to insure accuracy it is desirable that the sun 
should be on the prime vertical, or as near it as practicable. 
It should NOT he near the meridian, for at that time the sun 
is rising or falling very slowly, and the slightest error in the 
measured altitude would make an enormous difference in the 
computed hour angle. If the sun is exactly east or west at 
the time of observation, an error of even several minutes of 
arc in the assumed latitude produces no sensible effect upon 
the result. 


The disadvantage of the method is that any error of gradua- Disadvan- 
tion of the sextant vitiates the result, and no sextant is perfect. *^^® ^^ ^^^^ 

method and 

But with ordinary care and good instruments the sea-captain is umit of 
able to get his time correct within three or four seconds. accuracy. 

When a number of altitude observations have been made for time, and 
it is desired to reduce them separately, so as to test their agreement and 
determine their probable error, there is an advantage in using the formula 

cos L 

COS P = ^ — -- — tan (j) tan 8, 

cos (f> COS 6 

employing the " Gaussian logarithms " in the computation. The second 
term of the formula and the denominator of the first term remain constant 
through the whole series, saving much labor in reduction. 

104. To compute the Time of Sunrise or Sunset. — To solve calculation 
this problem we have precisely the same data as in finding the ^^ ^"^^® ^^ 

. -IT PI sunrise aud 

time by a smgle altitude ot the sun. The zenith-distance oi the sunset, 
sun's center at the moment when its upper edge is rising equals 
90° 51', — made up of 90° plus 16' (the mean semidiameter of 
the sun) plus 35' (the mean refraction at the horizon). The 
resulting hour angle, corrected for the equation of time, gives 
the mean local time at which the sun's upper limb reaches the 
horizon under average circumstances of temperature and baro- 
metric pressure. If the sun rises or sets over the sea horizon 
and the observer's eye is at any considerable elevation above 
sea-level, the dip of the horizo7i must also be added to 90° 51' 
before making the computation. 

The beginning and end of twilight may be computed in the 
same way by merely substituting 108°, i.e., 90° + 18°, for 
90° 51'. 


Having now the means of finding the true local time at any 
place, we can take up the problem of the longitude, the most 
important of all the economic problems of astronomy. The great 
observatories at Greenwich and Paris were established expressly 



Definition of 

of longitude 
equals dif- 
ference of 
local times. 

The knot 
of the 


Details of 

for the purpose of furnishing the observations which could be 
utilized for its accurate determination at sea. 

105. The longitude of a place on the earth may be defined as the 
angle at the pole of the earth hetiveen the standard meridian and the 
meridian of the place; and this angle is measured by, and equal 
to, the arc of the equator intercepted between the two meridiajis. 

As to the standard meridian there is some variation of usage. 
At sea nearly all nations at present reckon from the meridian of 
Greenwich, except the French, who insist on Paris. 

Since the earth turns on its axis at a uniform rate, the angle 
at the pole is strictly proportional to the time required for the 
earth to turn through that angle ; so that longitude may be, and 
now usually is, expressed in time units^ — z.e., in hours, minutes, 
and seconds, rather than degrees, etc., — and is simply the differ- 
ence between the local times at G-reenwich and at the place where 
the longitude is to he determined. 

Since the observer can determine his own local time by the 
methods already given, the knot of the problem is to find the 
Greenwich local time corresponding to his own, without leaving 
his place. 

106. First Method: by Telegraphic Comparison between his 
Own Clock and that of Some Station whose Longitude from Green- 
wich is known. — The difference between the two clocks will 
be the difference of longitude between the two stations after 
the proper corrections for clock errors^ persoyial equation., and 
time occupied hy the transmission of the electric signals have 
been applied or eliminated. 

The process usually employed is as follows : The observers, after ascer- 
taining that they both have clear weather, proceed early in the evening 
to determine the local time at each station by an extensive series of star 
observations with the transit-instrument. Then at an hour agreed upon 
the observer at the eastern station. A, " switches his clock " into the tele- 
graphic circuit, so that its beats are communicated along the line and 
received upon the chronograph of the western station. After the eastern 
clock has thus sent its signals, say for two minutes, it is '' switched out " 


and the luestern observer puts his clock into the circuit, so that its beats 
are received upon the eastern chronograph. Sometimes the signals are 
communicated both ways simultaneously, so that the beats of both clocks 
appear upon both chronograph sheets at the same time. The operation is 
closed by another series of transit observations by each observer. 

We have now upon each chronometer sheet an accurate comparison of the 
two clocks, showing the amount by which the western clock is slow of the 
eastern, and if the transmission of electric signals were instantaneous, 
the difference shown upon the two chronometer sheets would be identical 
on both. Practically, however, there will always be a discrepancy of some Elimmation 
hundredths of a second, amounting to twice the time occupied in the trans- of error due 
mission of the signals; but the mean of the two differences after correctincj o^^J^"^" 
for the carefully determined clock errors will be the true difference of longi- ..' , 
tude between the places. Especial care must be taken to determine with personal 
accuracy the personal equations of the observers, or else to eliminate them, which equation. 
may be done by causing the observers to change places. 

In cases where the highest accuracy is required, it is customary to make 
observations of this kind on not less than five or six evenings. 

The astronomical difference of longitude between two places Limit of 
can thus be determined within about ^V of ^ second of time, ^.e., attainable 

y . accuracy. 

within about 20 feet in the latitude of the United States. 

107. Second Method : by the Chronometer. — This method is The ciirono- 
available at sea. The chronometer is set to indicate Greenwich i^^etric 
time before the ship leaves port, its " rate " having been care- determining 
fully determined by observation for several- days. In order to longitude 
find the longitude by the chronometer, the sailor must determine 
its " error " upon local time by an observation of the altitude of 
the sun when near the prime vertical (Sec. 103). If the chro- 
nometer indicates true Greenwich time, its "error" deduced 
from the observation will be the longitude. Usually, however, 
the indication of the chronometer face must be corrected for 
the gain or loss of the chronometer since leaving poTt, in order 
to give the true Greenwich time at the moment. 

Chronometers are only imperfect instruments, and it is cinonome- 
important, therefore, that several of them should be carried by ters needed 
the vessel to check each other. This requires three at least, ^^y^\i 



Failure of 
the method 
for long 

method, the 
moon being 
regarded as 
a clock hand 

on land. 

at sea. 

because if only two chronometers are carried, and they disagree, 
there is nothing to indicate which is the delinquent. 

Moreover, in the course of months, chronometers generally 
change their rates progressively^ so that they cannot be depended 
on for very long intervals of time ; and the error accumulates 
much more rapidly than in proportion to the time. If, there- 
fore, a ship is to be at sea more than three or four months 
without making port, the method becomes untrustworthy. 
For voyages of less than a month it is now practically all 
that could be desired. 

108. Third Method : by the Moon regarded as a Clock Hand, 
with Stars for Dial Figures. — Before the days of reliable chro- 
nometers, navigators and astronomers were generally obliged to 
depend upon the moon for their Greenwich time. The laws 
of her motion are now fairly well known, so that the right 
ascension and declination of the moon are now computed and 
published in the Nautical Almanac, three years in advance, for 
every Greenwich hour of every day in the year. It is therefore 
possible to deduce the Greenwich time at any moment when 
the moon is visible by making some observation which will 
accurately determine her place among the stars. 

On land it may be : 

(a) The direct transit-instrument observation of her right 
ascension as she crosses the meridian. 

{h) The observation at the moment when she occults a star 
(incomparably the most accurate of all lunar methods) or makes 
contact with the sun in a solar eclipse. 

(e) The observation of the moon^s azimuth with the universal 
instrument at an accurately determined time. 

At sea the only practicable observation is to measure with 
a sextant a lunar distance^ i.e.^ the distance of the moon from 
some star or planet nearly in her path. 

Since, however, the almanac place of the moon is the place she would 
apparently occupy if seen from the center of the earth, most lunar 


observations require complicated and laborious reductions before they can Inferiority- 
be used for longitude. Moreover, the motion of the moon is so slow (she of the lunar 

requires a month to make the circuit of 360°) that any error in the "^^^^^ods, 

1 X- -c 1 1 J 1 J.1,- i. J.- ^ . occultations 

observation oi her place produces nearly thirty times as great an error m . + i 

the corresponding Greenwich time and the deduced longitude. It is as if 
one should try to read accm-ate time from a watch that had only an hour- 

109. Other Methods: Eclipses of the Moon and Jupiter's Satellites. — Longitude 
A rough longitude can be obtained from the observation of these eclipses, by eclipses 
since they occur at the same moment of absolute time wherever observed. * ® moon 
By comparing the local times of observation with the Greenwich time -, . ,, 
obtained by correspondence after the event, or from the Almanac, the satellites, 
difference of longitude at once comes out. The difficulty with this method 

is that the eclipses are gradual phenomena, presenting no well-marked 
instant for observation. 

On the same principle art'ificial signals, such as flashes of powder and Longitude 
explosion of rockets, can be used between two stations so situated that ^J artificial 
both can see the flashes. Early in the century the difference of longitude ^ 
between the Black Sea and the Atlantic w^as determined by means of a 
chain of such signal stations on the mountain tops ; so also, later, the differ- 
ence of longitude between the eastern and western extremities of the 
northern boundary of Mexico. This method is now superseded by the 

110. Local and Standard Time. — Until recently it has been Local and 
always customary to use local time, each city determining its ^.^" 
own time by its own observations. Before the days of the 
telegraph, and while traveling was comparatively slow and infre- 
quent, this was best. At present it has been found better for 

many reasons to give up the system of local times in favor of a 
system of standard time. This facilitates all railway and tele- Advantages 
graphic business in a remarkable degree, and makes it practi- ^^ standar 
cally easy for every one to keep accurate time, since it can be 
daily wired from some observatory (as Washington) to every 
telegraph office in the country. According to the system now American 
established in North America, there are five such standard times ^5^**1^"^,°^ 

' ^ standard 

in use, — the colonial^ the eastern^ the central., the mountain., time. 
and the Pacific, — which are slower than Greenwich time by 




exactly four, five, six, seven, and eight hours, respectively. 
The minutes and seconds are everywhere identical. 

At most places only one of these standard times is employed ; 
but in cities where different systems join each other, as, for 
instance, at Atlanta and Pittsburg, two standard times are in 
use, differing from each other by exactly one hour, and from 
the local time by about half an hour. In some such places the 
local time also maintains itself. 

The same system has now been practically adopted in almost 
all civilized countries. 

time in 

On the continent of Europe, except in Russia, the time used is one hour 
fast from Greenwich ; in West Australia, eight hours fast ; in Japan and 
South Australia, nine hours ; in Victoria, Queensland, and Tasmania, ten 
hours. Cape Colony uses time one and one-half hours fast from Greenwich 
time, and New Zealand eleven and one-half hours fast. 

The begin- 
ning of the 
day at the 
180th me- 

In order to determine the standard time by observation it 
is only necessary to determine the local time by one of the 
methods given, correcting it by first adding the observer's longi- 
tude west from Greenwich, and then deducting the necessary 
integral number of hours. 

111. Where the Day begins. — It is evident that if a traveler 
were to start from Greenwich on Monday noon and were by 
some means able to travel westward along the parallel of lati- 
tude as fast as the earth turns eastward beneath his feet, he 
would keep the sun exactly upon the meridian all day long 
and have continual noon. But what noon? It was Monday 
noon when he started, and when he gets back to London 
twenty-four hours later he will find it to be Tuesday noon there. 
Yet it has been noon all the time. When did Monday noon 
become Tuesday noon ? 

It is agreed among mariners to make the change of date at the 
180th meridian from Grreenwich, which passes over the Pacific 
hardly anywhere touching the land. 


Ships crossing this line fro7n the east skip one day in so i.oss or gain 
doingf. If it is Monday afternoon when a ship reaches the *' ^ ^^^ ^^^ 

°^ '^ ^ vessels pass- 

line, it becomes Tuesday afternoon the moment she passes it, ingthe 

the intervening twenty-four hours being dropped from the ^^^te-lme. 

reckoning on the log-book. Vice versa^ when a vessel crosses 

the line from the western side, it counts the same day twice 

over^ passing from Tuesday back to Monday, and having to do 

Tuesday over again. 

There is considerable irregularity in the date actually used on the The date- 
different islands in the Pacific, as will be seen by looking at the so-called ^^^^' 
date-line as given in the Century Atlas of the World. Those islands which 
received their earliest European inhabitants via the Cape of Good Hope have 
adopted the Asiatic date, even if they really lie east of the 180th meridian ; 
while those that were first approached via the American side have the 
American date. When Alaska was transferred from Russia to the United 
States, it was necessary to drop one day of the week from the official dates. 


112. Determination of the Position of a Ship. — The determi- 
nation of the place of a ship at sea is commercially of such 
importance that, notwithstanding a little repetition, we collect 
here the different methods available for the purpose. They 
are necessarily such that the requisite observations can be 
made with the sextant and chronometer, the only instruments 
available on shipboard. 

The latitude is usually obtained by observations of the sun's Determina- 
altitude at noon, according" to the method explained in Sec. 90. *^<^^oi'^ 

\ ° . ^ ship's lati- 

The longitude is usually found by determining the error upon tude and 
local time of the chronometer, which carries Greenwich time, lo^^gitude. 
(See Sees. 103 and 107.) 

In case of long voyages, or when the chronometer has for 
any reason failed, the longitude may also be obtained by meas- 
uring lunar distances and comparing them with the data of the 
Nautical Almanac. 




The circle 
of position. 
Its center 
and radius 
at any time. 

Position of 
ship deter- 
mined by 
the inter- 
section of 
two circles 
of position. 

These methods require separate observations for the latitude 
and for the longitude. 

113. Sumner's Method. — At present a method known as 
Sumner s Method^ because first proposed by Captain Sumner of 
Boston, in 1843, has come largely into use. It is based on the 
principle that any single observation of the sun's altitude, giving, 
of course, its zenith-distance at the time, determines the so-called 
circle of position on which the ship is situated. The center of 
this circle of position on the earth's surface is the point directly 
under the sun at the moment of observation. The longitude of 
this point is the Grreenwich apparent time at the moment of 
observation as determined by the chronometer, and its latitude 
is the sun's declination. The radius of the circle of position 
(reckoned in degrees of a great circle from this center) is the 
observed zenith-distance of the sun. 

A second observation made some hours later will give a 
second circle of position, and if the ship has not moved mean- 
while the intersection of the two circles will give the place of 
the ship. 

The circles intersect at two points, of course, but at which 
one the ship is situated is never doubtful, because the approxi- 
mate azimuth of the sun, observed simply as a compass bearing, 
tells roughly on what part of the circle the ship is placed. 
If, for instance, the sun is in the southeast at the first observa- 
tion, the ship must be on the northwestern part of the corre- 
sponding circle of position. 

If the ship has moved between the two observations, as of 
course is usual, its motion as determined by log and compass 
can be allowed for with very little difficulty. 

of the 

114. Usually the matter is treated as follows : The latitude of the 
vessel is practically always known within a degree or so, from the " dead 
reckoning " since the last observation. Suppose the latitude is known to 
be about 51° ; then, from the first (morning) observation of the sun's 
altitude and the chronometer time, the navigator computes the longitude, 



assuming the latitude, to he 50°, and finds it to be, say, 40° 52'. Again, 
assuming the latitude to be 52°, he gets 43° 20', and marks the two com- 
puted longitudes at A and B on the chart (Fig. 39). A line drawn through 
these points will be very nearly a part of the vessel's circle of position at 
the time of that observation. 

From the second (afternoon) observation the points C and D are 
computed in the same way, giving a piece of the second circle of position. 

Suppose now that in the interval the ship has moved 60 miles on a 
course north 60° west. From the points A and B lay off 60 miles on the 

Longitude West from Greenwich 

chart in the proper 
direction to the points 
a and b, and join ab 
by a line. aS", the in- 
tersection of this line 
with the line CD, will 
be the position of the 
ship at the time of 
the second observation 
with all the approxi- 
mation necessary for 
the navigator's pur- 
pose ; and if we reckon 
back 60 miles from S', 
we shall find S, the 
ship's position in the 
morning. There are, 

however, extended tables which greatly reduce the labor of computations 
and make the result more accurate than that derived from the chart. 

The peculiar advantage of the method is that each observa- 
tion is used for all it is worth, giving accurately the position its peculiar 
of a line upon which the vessel is somewhere situated, and ^ vantage, 
approximately (by the sun's azimuth) its position on that line. 
Very often this knowledge is all that the navigator needs to 
give him the knowledge of his distance from land, even when 
he fails in getting the second observation necessary to deter- 
mine his precise location. Everything^ however^ depends upon Must have 
the correctness of the G-reenwich tiyne given hy the chronometer^ Gi-eeuwicii 

just as in the ordinary method of longitude determination. 
L. of C. 

Fig. 39. — Sumner's Method 



tion of 
azimuth by 
of pole-star. 

115. Determination of *' Azimuth." ^ — A problem, important, though 
not so often encountered as that of latitude and longitude determinations, 
is that of determining the "azimuth," or "true bearing," of a line upon 
the earth's surface. 

With a theodolite having an accurately graduated horizontal circle the 
observer points alternately upon the pole-star and upon a distant signal 
erected for the purpose at a distance of say half a mile or more, — usually 
an " artificial star " consisting of a small hole in a plate of metal, with 
a lantern behind it. At each pointing he notes the time by a sidereal 
chronometer. The theodolite must be carefully adjusted for collimation, 

and especial pains must be taken to have the 
axis of the telescope perfectly level. 

The next morning by daylight the observer 
measures the angle or angles betv^een the 
night signal and the objects vi^hose azimuth 
is required. 

If the pole-star were exactly at the pole, 
the mere difference between the two readings 
of the circle, obtained when the telescope is 
pointed on the star and on the signal, would 
directly give the azimuth of the signal. As 
this is not the case, the azimuth of the star 
must be computed for the moment of each 
observation, which is easily done, as the 
right ascension and declination of the star 
are given in the Almanac for every day of the year. 

Referring to Fig. 40, N being the north point of the horizon, P the pole, 
and NZ the meridian, we see that PS is the polar distance of the star, or 
complement of its declination, the side PZ is the complement of the 
observer's latitude, while the angle at P is the hour angle of the star, i.e., 
the difference between the right ascension of the star and the sidereal time 
of observation. This hour angle must, of course, be reduced to degrees 
before making the computation. We thus have two sides of the triangle, 
viz., PS and PZ, with the included angle at P, from which to compute 
the angle Z at the zenith. This is the star's azimuth. 

The pole-star is used rather than any other because, being so near 
th« pole, any slight error in the assumed latitude of the place or in the 
time of the observation will produce hardly any effect upon the result, 
especially if the star be observed near its greatest elongation east or west 
of the pole. 

Fig. 40. 

N H H' 

- Determination of 


The sun, or any other heavenly body whose position is given in the Azimuth by 
Almanac, can also be used as a reference point in the same way when near the sun. 
the horizon, provided sufficient care is taken to secure an accurate observa- 
tion of the time at the instant when the pointing is made. But the results 
are usually rough compared with those obtained from the pole-star. 


116. The "position" of a heavenly body is defined by its 
right ascension and declination. These may be determined : 

(1) By the meridian-circle., provided the body is bright enough Direct deter- 
to be seen by the instrument and crosses the meridian at """^^lon of 

both coord i- 

night. If the instrument is in exact adjustment, the sidereal nates of a 
time when the object crosses the middle ivire of the reticle of body's posi- 
the instrument is directly the right ascension of the object, nieridian- 
Corrections are necessary only on account of errors of the clock, fii'^'ie 
errors of adjustment of the instrument, and personal equation 
of the observer. Parallax and refraction do not enter into the 

The reading of the circle of the instrument, corrected for 
refraction, and for parallax if necessary, gives the polar distance 
of the object if the polar point of the circle has been deter- 
mined, or it gives the zenith-distance of the object if the 
nadir point has been determined (Sec. 69). In either case the 
declination can be immediately deduced. A single complete 
observation, therefore, with the meridian-circle, determines both 
the right ascension and declination of the object. In order to 
secure accuracy, however, it is desirable that the observations 
should be repeated many times. 

It is often better to use the instrument " differentially," i.e., Differential 
to observe some neighboring standard star or stars of accurately "'^^ °^ ^^^® 

1 •• in fit' 1 • iweridian- 

xnown position, soon beiore or aiter the object whose place is eircie. 
to be determined. We thus obtain the difference between the 
right ascension and declination of the object observed and others 



which are accurately known; and in this case slight errors in 
the graduation and adjustment of the instrument affect the final 
result very little. 

tion of posi- 
tion by the 
and microm- 

When a body (a comet, for instance) is too faint to be observed 
by the telescope of the meridian-circle, which is seldom very 
powerful, or when it does not come to the meridian during the 
night, we must accomplish our observation with some instrument 
that can pursue the object to any part of the heavens. At 
present the equatorial is almost exclusively used for the purpose. 

117. (2) By the equatorial. With this we determine the posi- 
tion of a body by measuring the difference of right ascension and 
declination between it and some neighboring star whose place 
is given in a star-catalogue, and, of course, has been accurately 
determined by the meridian-circle of some observatory. 

In measuring this difference of right ascension and declination 
we usually employ a micrometer (Sec. 71) fitted with wires like 
the reticle of a meridian-circle. It carries a number of fixed 
wires which are set accurately north and south in the field of 
view, and these are crossed at right angles by one or more wires 
which can be moved by the micrometer screw. The difference 
of right ascension between the star and the object to be deter- 
mined is measured by clamping the telescope firmly and simply 
observing and recording upon the chronograph the transits of 
the two objects across the wires that run north and south; the 
difference of declination, by bisecting each object by one of 
the micrometer wires as it crosses the middle of the field of 
view. The difference of the two micrometer readings gives the 
difference of declination. 

The observed differences must be corrected for refraction 
and for the motion of the body during the time of observation. 

The measurement may also be made with the position microm- 
eter by measuring the angle of position and distance between 
the object and the star of comparison, as it is called. 


Instead of using a micrometer we may employ photography. Determina- 
For this purpose the telescope is fitted with a plate-holder in ^^^^] °^ 

, . . IDOSitlou by 

place of the eyepiece, and is accurately driven by clockwork, i^hotog- 
On the sensitive plate a photograph is obtained of all the stars ^M^y- 
in the field, and also of the object; and the position of the 
object is afterwards determined by measuring the plate. It is 
found that determinations of extreme accuracy can be made in 
this way, and the method is rapidly coming into extensive use. 


In cases where corrections for refraction are required they are to be 
taken from Table VIII (Appendix), taking into account the temperature 
and barometric pressure, if given among the data. If preferred, the student 
may also use Comstock's formula (Sec. 82). The results for example 1 
have their corrections computed by the regular refraction tables, and the 
approximate results obtained by the student may differ from them by a 
considerable fraction of a second. 

1. Given the following meridian-circle observations on /3 Ursse Minoris 
at its upper and lower culminations, respectively, viz. : 

Altitude 55° 48' 06''.0, temperature 30° F., barometer 30.1 inches. 
24° 58' 56''.4, " 25° F., " 30.1 « 

The nadir reading (Sec. 69) was 270° 01' 06".8 in both cases. Required 
the latitude of the place and the declination of the star. 

Ans. Lat. 40° 20' 57". 8. 
Dec. 74° 34' 40".l. 

2. Given the meridian altitude of the sun's lower limb 62° 24' 45", the 
height of the observer's eye above the sea-level being 16 feet (Sec. 77). 

The sun's declination was + 20° 55' 10" and its semidiameter 15' 47". 
Its parallax at the observed altitude was 5" and the mean refraction from 
Table YIII may be used. Required the latitude of the ship. 

Ans. + 48° 19' 3". 

3. The sun's meridian altitude on a ship at sea is observed to be 30° 15' 
(after being duly corrected); the sun's declination at the time is 19° 25' 
south. What is the ship's latitude ? 

4. How much will a sidereal clock gain on a mean solar clock in 10 
hours and 30 minutes? 

Ans. 1^43.53. 


5. How many times will the second-hand of a sidereal clock overtake 
that of a solar clock in a solar day if they start together ? 

Ans. 236 times. 

6. At what intervals do the coincidences occur? 

Ans. 6™5.242^ 

7. Reduce 10 hours 40 minutes and 25 seconds of mean time to 
sidereal time. (See Sec. 99.) 

8. Reduce 10 hours 40 minutes and 25 seconds of sidereal time to 
solar time. 

9. What is the approximate sidereal time on July 30 at 10 p.m. ? 

Solution hy note to Sec. 99. July 22, noon sid. time = 8^00™ 

8 days gain 32 

Sid. time at noon 8^32™ 

10 hours = sid. 10 If 

Sid. time at 10 p.m. 18^33^™ 

10. What is the approximate sidereal time on October 4 at 7 a.m. 
civil reckoning ? 

11. In determining longitudes by telegraph, will it or will it not make 
any difference whether sidereal or solar clocks are used by the observers, 
provided both use the same ? 

12. A ship leaving San Francisco on Tuesday morning, October 12, 
reaches Yokohama after a passage of exactly 16 days. On what day of 
the month and of the week does she arrive ? 

13. Returning, the same vessel leaves Yokohama on Saturday, Novem- 
ber 6, and reaches San Francisco on Tuesday, November 23. How many 
days was she on the voyage ? 


Its Form, Rotation, and Dimensions — Mass, Weight, and Gravitation — The Earth's 

Mass and Density 

118. In a science which deals with the heavenly bodies it 
might seem at first that the earth has no place ; but certain 
facts relating to it are similar to those we have to study in 

the case of sister planets, are ascertained by astronomical The earth an 
methods, and a knowledge of them is essential as a basis of all astrononn- 
astronomical observations. In fact astronomy, like charity, j^ many 
" begins at home," and it is impossible to go far in the study respects. 
of the bodies which are strictly " celestial " until one has 
acquired some accurate knowledge of the dimensions and 
motions of the earth itself. 

119. The astronomical facts relating to the earth are broadly 
these : 

(1) The earth is a great hall about 7920 miles i^i diameter. Leading 

(2) It rotates on its axis once in ttventy-four sidereal hours. astronomi- 
^ ' ^ . , (^al facts 

(3) It is not exactly spherical.^ hut is flattened at the poles, the relating to 

polar diameter heing nearly 27 miles, or prohahly a little more **^® earth. 
than one three-hundredth part less than the equatorial. 

(4) Its mean density is between 5.5 and 5.6 as great as that of 
water, and its mass is represented in tons by 6 with twenty-one 
ciphers following {six thousand millions of m,illions of millions 
of tons). 

(5) It is flying through space in its orbit around the sun with 
a velocity of about 18^ miles a second, or nearly 100000 feet a 
second, — about thirty -three times as fast as the swiftest modern 




of the earth ; 
its shadow 

tion of its 
diameter by 
an arc of 


120. The Earth's Approximate Form. — It is not necessary to 
dwell on the ordinary familiar proofs of the earth's globularity. 
One, first quoted by Galileo as absolutely conclusive, is that 
the outline of the earth's shadow seen upon the moon during a 
lunar eclipse is such as only a sphere could cast. 

We may add, as to the smoothness and roundness of the 

earth, that if represented by an 18- 
inch globe, the difference between 
its greatest and least diameters would 
be only about Jg ^^ ^^ inch, the 
highest mountains would project only 
about Jg of an inch, and the average 
elevation of continents and depths of 
the ocean would be hardly greater 
on that scale than the thickness of a 
film of varnish. Relatively, the earth 
is much smoother and rounder than 
most of the balls in a bowling-alley. 
121. The Approximate Measure of 
the Diameter of the Earth regarded as 
a Sphere. — (1) By an arc of merid- 
ian. There are various ways of deter- 
mining the diameter of the earth. 
The simplest and best is by measuring 
the length of an arc. It consists essentially in astronomical 
measurements which determine the distance between two selected 
stations (several hundred miles apart) in degrees of the earth's 
circumference, combined with geodetic measurements giving 
their exact distance in miles or kilometers. 

The astronomical determination is most easily made if the 
two stations are on the same terrestrial meridian. Then, as is 
clear from Fig. 41, the distance ah in degrees is simply the 

Fig. 41. — Measuring the Earth's 



difference of latitude between a and h. The latitudes are best 
determined by zenith-telescope observations (Sec. 92), but any 
accurate method may be used. 

The linear distance (in feet or meters) is measured by a 
geodetic process called triangulation. It is not practicable to 
measure it with sufficient accuracy directly, 
as by simple " chaining." 

Between the two terminal stations (^A 
and J7, Fig. 42) others are selected, such 
that the lines joining them form a com- 
plete chain of triangles, each station being B, 
visible from at least two others. The 
angles at each station are carefully meas- 
ured, and the length of one of the sides, b 
called the hase^ is also measured with all 
possible precision. It can be done, and is 
done, with an error not exceeding an inch 
in 10 miles. {B U is the base in the figure.) 
Having the length of the base and all the 
angles, it is then possible to calculate every 
other line in the chain of triangles and to 
deduce the exact north and south distance 
{Ha) between H and A. An error of more 
than three feet in a hundred miles would 
be unpardonable. 

In this way many arcs of meridian have 
been measured the average of which (for they differ, because 
the earth, as we shall see, is not quite spherical) makes the 
length of a degree 69.1 miles, the mean circumference 24875 
miles, and the mean diameter 7918 miles. 

122. The ancients understood the principle of the operation 
perfectly. Their best known attempt at a measurement of the 
sort was made by Eratosthenes of Alexandria about 250 B.C., 
his two stations being Alexandria and Syene in Upper Egypt. 

Fig. 42. — Triangulation 

ical work 
consists in 
the latitudes 
of the termi- 
nal stations. 


consists in 
distance in 
miles by 




Method of 

At Syeiie he observed that at noon of the longest day in summer 
there was no shadow in the bottom of a well, the sun being 
then vertically overhead. On the other hand, the gnomon at 
Alexandria on the same day, by the length of the shadow, gave 
him Jq- of a circumference (7° 15') as the distance of the sun 
from the zenith at that place. This -^-^ of a circumference is, 
therefore, the difference of latitude between Alexandria and 
Syene, and the circumference of the earth must be fifty times 
the linear distance between those two stations. 


Fig. 43. — Curvature of the Earth's Surface 

The weak place in his work was the measurement of this linear distance 
between the two places. He states it as 5000 stadia, without telling how it 
was measured, thus making the circumference of the earth 250000 stadia, 
which may be exactly right ; for we do not know the length of his stadium, 

nor does he give any 

— . Y account of the means 

by which he measured 
the distance, if he meas- 
ured it at all. (There 
seem to have been as 
many different stadia 
among the ancient na- 
tions as there were kinds of " feet " in Europe at the beginning of the 

The first really valuable measure of an arc of the meridian was that 
made by Picard in Northern France in 1671, — the measure which served 
Newton so well in his verification of the idea of gravitation. 

Experi- 123. (2) The curvature of the earth's surface is easily demon- 

mentai strated, and an approximate estimate of its diameter obtained, 

exhibition ^ ^ 

of curvature by the following method. Erect upon a reasonably level plane 
of earth's three rods in line, a mile apart, and cut off their tops at the 

surface. ptit -tt ^ t t • , 

same level, carefully determined by a surveyor s leveimg-mstru- 
ment. It will then be found that the line AC (Fig. 43), joining 
the extremities of the two terminal rods, when corrected for 
refraction, passes about 8 inches below B, the top of the 
middle rod. 


(On account of refraction, however, which curves the line of Effect of 
sis-ht between A and C, the result cannot be made exact. The y^^^^^tion m 

o lessening 

observed value of BD ranges all the way from 4.5 to Q.b inches, the apparent 
according to the temperature.) curvature. 

Suppose the circle ABC completed, and that E is the jjoint 
on the circumference opposite B^ so that BE equals the diameter 
of the earth {=2B). 

By geometry, BB : BA = BA : BE, or 2 E, Computa- 

^ 42 t^on of 

whence ^ = ---. e^^th'^ 

2j JijJ diameter 

from such 

Now BA is 1 mile, and BI) = 8 inches, or | of a foot, or y 9^2 observa- 
of a mile. ^ *^^"'- 

Hence, 2 B (the earth's diameter) = 7920 miles, — a very fair 


124. Probable Evidence of the Earth's Rotation. — At the time 
of Copernicus the only argument he could bring in favor of 
the earth's rotation was that this hypothesis was much more 
probable than the older one that the great heavens themselves 
revolved. All the phenomena then known would be sensibly 
the same on either supposition. The apparent diurnal motion Probability 
of the heavenly bodies can be fully accounted for within the ^^^^^^ 

•^ . . -^ earth's 

limits of observation then possible, either by supposing tliat rotation. 
the stars are actually attached to an immense celestial sphere 
which turns around daily, or that the earth itself rotates upon 
an axis ; and for a long time the latter hypothesis seemed to 
most people less probable than the older and more obvious one. 

A little later, after the invention of the telescope, analog?/ 
could be adduced ; for with the telescope we can see that the 
sun, moon, and many of the planets are rotating globes. 

Within the last century it has become possible to adduce 
experimental proofs which go still further and absolutely 



proof of 
the earth's 
rotation by 
the Foucault 

demonstrate the earth's rotation. Some of them even make 
it visible. 

125. Foucault^s Pendulum Experiment. — Among these experi- 
mental proofs the most impressive is the pendulum experiment, 
devised and first executed by Foucault in 1851. From the 
dome of the Pantheon in Paris he hung a heavy iron ball about 
a foot in diameter by a wire more than 200 feet long (Fig. 44). 
-_^- A circular rail some 12 feet 

across, with a little ridge of sand 
built upon it, was placed in such 
a way that a pin attached to the 
swinging ball would just scrape 
the sand and leave a mark at 
each vibration. To put the ball 
in motion it was drawn aside by 
a cotton cord and left for hours 
to come absolutely to rest ; then 
the cord was burned and the 
pendulum started without jar to 
SAving in a true plane. 

But this plane at once began 
apparently to deviate slowly to- 
ivards the right, in the direction 
of the hands of a watch, and 
the pin on the pendulum ball cut 
the sand ridge in a new place 
at each swing, shifting at a rate which would carry it completely 
around in about thirty-two hours if the pendulum did not first 
come to rest. In fact, the floor of the Pantheon was really and 
visibly turning under the plane of the vibrating pendulum. 

Fig. 45 is copied from a newspaper of that date and shows 
the actual appearance of the apparatus and its surroundings. 
The experiment created great enthusiasm at the time and has 
since been frequently repeated. 

Fig. 44. — Foucault's Pendulum 



KbMi I* te^rxw m-t>,w4 voMtftt- 

4fm iwtMiHCjHMHNwMMHMi 

> H«K. Hot* *» XMtcW •!•} 

126. Explanation of the Foucault Experiment. — The approxi- 
mate theory of the experiment is very simple. A swinging 
pendulum, suspended so as to be equally/ free to siving in any 
plane (unlike the common clock pendulum in this freedom), if 
set up at the pole of the earth, would appear to shift 'completely 
around in twenty-four hours. Really in this case the plane of 
vibration remains unchanged while the earth turns under it. 
This can easily be understood by setting up on a table a similar 
apparatus, consisting- of a ETr^nlF^^™"^ ': ^ -^i^-" 

ball hung from a frame by 
a thread, and then, while 
the ball is swinging, turn- 
ing the table around upon 
its casters with as little jar 

as possible. The plane of 
the swing will remain un- 
changed by the motion of 
the table. 

It is easy to see, more- 
over, that at the earth's 
equator there will be no 
such tendency to shift; 
while in any other latitude 
the effect will be interme- 
diate and the time for the 

pendulum to complete the revolution in its plane will be longer 
than at the pole. 

The northern edge of the floor of a room in the northern 
hemisphere is nearer the axis of the earth than is its southern 
edge, and therefore is carried more slowly eastward by the 
earth's rotation. Hence, the floor must " skew " around con- 
tinually, like a postage-stamp gummed upon a whirling globe, 
anywhere except at the globe's equator. A line drawn on 
the floor, therefore, continually shifts its direction, and a free 

Fig. 45. — Foucault's Pendulum in the 

Why the 
free pendu- 
lum appears 
to shift its 
plane. Rate 
of shift at 
the i^ole. 

No shift at 
the equator, 

Effect at 



15° X siu</). 

the rotation 
of the earth. 

pendulum, set at first to swing along such a line, must appar- 
ently deviate at the same rate in the opposite direction. 

It can be proved (see General Astro7iomt/, Arts. 140, 141) 
that the hourly deviation of a Foucault pendulum equals 15° 
multiplied by the sifie of the latitude. In the latitude of New 
York it is not quite 10° an hour. 

In the northern hemisphere the plane of vibration, as already 
stated, moves around with the hands of a watch. In the 
southern the motion is reversed. 

127. There are various other demonstrations of the earth's 
rotation which we merely mention, referring to the author's 
General Astronomy for their discussion : 

(a) By the gyroscope, an experiment also due to Foucault. 

(h) By the slight eastward deviation of bodies in falling from a 
height. This deviation is, of course, zero at the pole and a maxi- 
mum at the equator ; it varies as the cosine of the latitude, other 
things being equal, and amounts to about one inch in a fall of 
500 feet for a station in latitude 50°. The idea of the experi- 
ment is due to Newton, but its execution has been carried out 
only during the past century, by several European observers. 

If the earth were strictly spherical, and if gravity were 
directed to its center, there would also be a slight deviation 
towards the equator. But Laplace has shown that, as things 
are, no sensible deviation of that kind takes place. 

(c) By the deviation of projectiles to the right in the northern 
hemisphere, to the left in the southern. 

(d) By various phenomena of meteorology and physical geog- 
raphy, — such as the direction of the trade and anti-trade- 
winds and of the great ocean currents, and the counter-clockwise 
revolution of cyclones in the northern hemisphere, reversed in 
the southern. 

It might at first seem that the rotation of the earth once a 
day is not a very rapid motion, but a point on the equator 
travels nearly 1000 miles an hour, or about 1500 feet a second. 


128. Invariability of the Earth's Rotation It is a question 

of great importance whether the day ever changes its length, Question as 
for if it does our time unit is not a constant. Theoretically, ^"jj^^j" 
some change is almost inevitable. The friction of the tides earth's 
and the deposit of meteoric matter on the earth's surface both notation. 
tend to retard its rotation ; on the other hand, the earth's loss 
of heat by radiation, and its consequent shrinkage, tend to 
accelerate it and to shorten the day. Then geological causes 
act, some one way and some the other. 

At present we can only say that the change which may have 
occurred since astronomy has been accurate is too small to be Certain that 
surely detected. The day is certainly not longer or shorter by ^'^^^§6' 'f 
_^_ part of a second than it was in the days of Ptolemy, and been 
probably it has not altered by i^-^-^ of a second. The test is extremely 
found in comparing the times at which celestial phenomena, 
such as eclipses, transits of Mercury, etc., have occurred during 
the range of astronomical history. 

Professor Newcomb's investigations in this line make it 
highly probable, however, that the length of the day has not Some indi- 
been quite constant during the last 150 years. There are cations of 
suspicious indications that Greenwich noon has, at irregular acceiera- 
intervals of from thirty to fifty years, sometimes come too early *^°^^ ^^^ 

r. J J ' ' P ^^ retarda- 

by as much as four or five seconds, and at other times fallen as tions. 
much behind. Astronomers are somewhat anxious on the sub- 
ject, because if the earth's rotation should turn out to be capri- 
ciously changeable in any sensible degree, it would compel us 
to look for some new and independent unit of time. 


129. Centrifugal Force due to the Earth's Rotation. — As the 
earth rotates on its axis every particle of its surface is sub- 
jected to a " centrifugal force " directed perpendicularly away 



force due 
to earth's 
rotation at 
the equator 
equals 559 of 

A rotation 
times as 
rapid would 
cause bodies 
to fly off 
from the 

force at any 
latitude is 
tional to the 
cosine of 
the latitude. 

from the axis, and this force, C, depends upon the velocity of 
the particle and the radius of the circle in which it moves. 
According to a familiar formula, this centrifugal force, C, 

equals --;- [Physics^ p. 74). An equivalent formula, often more 

4 ir^B 
convenient, is (7 = — , since V equals 2 irR, the circumfer- 
ence of the circle, divided by T, the time of revolution. This 
gives Cas an " acceleration " (in feet per second), just as gravity 
is given by g. 

The equatorial radius of the earth being 20 926202 feet, 
and the time of revolution, or the sidereal day, being 86167 

mean solar seconds, we find C = 0.1112 
feet per second, or 1.334 inches per 
second. This is ^\^ part of g^ which 
is 386 inches per second. At the 
earth's equator, therefore, C equals 

2 "so" 9' 

Since centrifugal force varies with 

the square of the velocity, and 289 
is the square of 17, it appears that if 
the earth revolved seventeen times as 
swiftly, keeping its present size and form (an impossible suppo- 
sition), bodies at the equator would lose all their weight; and 
if the speed were increased beyond that point, everything on 
the surface there would fly off unless fastened down. 

At any other latitude, assuming the earth to be spherical, 
which is sufficiently accurate for our present purpose, the 
radius of the circle described by M (Fig. 46) is MN^ which 
equals R cos (/> ; ^.e., at any latitude the centrifugal force c 
equals C cos (/> = 2\~^ 9 x ^^^ ^' becoming, therefore, zero at 
the pole. 

130. Effects of Centrifugal Force on Gravity. — At the equator 
the whole centrifugal force is opposed to gravity, so that bodies 

Fig. 46. — Centrifugal Force 
Caused by Earth's Rotation 



there weigh on this account ^ig^ less (weighed by a spring- 
balance) than they would if the earth did not rotate; but the 
direction of gravity is not altered. Elsewhere, except at the 
pole itself, both the amount and direction of gravity are 

(a) Diminution of G-ravity, Referring to the figure, we see 
that the centrifugal force MT^ or <?, is resolvable into two compo- 
nents, of which MR acts radially in direct opposition to gravity 
and equals MT x cos TMR = c cos cj) = C cos^ 0. 

(h) Change of Direction of G-ravity. MS., the other compo- 
nent of (?, acts horizontally towards the equator and equals 
C cos (f) sin ^ = \ C sin 2 <^. 
It acts to make the plumb-line 
hang away from the radius 
towards a point between 
and Q. 

In latitude 45° this hori- 
zontal force has its maximum 
and is about -^^g- part of 
the whole force of gravity, 
causing the plumb-line to 
deviate towards the equator 
about 11^ from the radius. 

If the earth's surface were spherical, this horizontal force 
would make every loose particle tend to slide towards the equa- 
tor, and the water of the ocean would so move. As things 
actually are, the surface of the earth has already arranged itself 
accordingly, and the earth bulges at the equator just enough to 
correct this sliding tendency. 

This effect of the earth's rotation on its form is well illustrated by the 
familiar little piece of apparatus shown in Fig. 47. 

If the earth's rotation were to cease, the Mississippi River would at once 
have its course reversed, since its mouth is several thousand feet farther 
from the center of the earth than are its sources. 

Fig. 47. 

Effect of Earth's Rotation on its 


into two 

opposed to 

direction of 

Maximum at 
45° latitude. 



gravity and 

tion of 
metliods for 
the form of 
the earth. 

Definition of 
or ellip- 

131. Gravity is not simply gravitation^ — the attraction of the 
earth for a body upon its surface, — but is the resultant of 
this attraction combined with the centrifugal force at the point 
of observation^ as above explained. It is this resultant force 
which determines the weight of a body at rest or its velocity 
and direction when falling. Only at the equator and poles is 
gravity directed towards the center of tlie earth. Surfaces of 
level are, on hydrostatic principles, necessarily everywhere per- 
pendicular to gravity, and are therefore not spheres around the 
earth's center, but spheroids flattened at the poles. 

132. The Earth's Form. — There are three ways of determin- 
ing the form of the earth : First, by geodetic measurement of 
distances upon its surface in connection with the astronomical 
determination of the points of observation. This gives not only 
the form but also the linear dimensions in miles or kilometers. 

Second, by observing the varying force of gravity at points 
in different latitudes, — observations which are made by means 
of a pendulum apparatus of some kind and determine only the 
form but not the size of the earth. 

Third, by means of purely astronomical phenomena., known as 
precession and nutation (to be treated of hereafter), and by 
certain irregularities in the motion of the moon. Observations 
of the occultations of stars at widely distant stations can also 
be utilized for the same purpose. These methods, like the 
pendulum method, give only the form of the earth. 

It is usual to characterize the form of the earth by its oblate- 
ness^ or ellipticity., though the latter term is rather objectionable 
on account of the danger of confounding it with eccentricity. 

The oblateness (O) is the fraction obtained by dividing the 
difference between the polar and equatorial semidiameters by 

A— B 

the equatorial, i.e.., fi = — — — • In the case of the earth this is 

about 3^1^^, but determinations by different methods range all 
the way from about -^\-^ to -^\r^. 



The eccentricity of an ellipse is 


1^ - /j^ 

and is always a much larger 

quantity than the oblateness, or ellipticity. In the case of the earth's 
meridian it is about t-jj- Its symbol in astronomy is usually e. 

J- "^ T 

133. Geodetic Method, by which Dimensions of the Earth as 
well as its Form are determined. — The method in its most 
convenient shape consists essentially in the measurement of 
two (or more) ares of meridian in widely different latitudes. 
These measurements are effected by the same combination of 
astronomical and geodetic operations already described for the 
measurement of a single arc (Sec. 121). More than twenty 
have thus far been meas- 
ured in various parts of <^\ ^ '^ 
the earth. The two 
longest are the great 
Kusso-Scandinavian arc, 
extending from Hammer- 
fest to the mouth of the 
Danube, and the Indian 
arc of practically equal 
length, reaching from the 
Himalayas to the south- 
ern extremity of the great peninsula. These are both between 
25° and 30° long; few of the others exceed 10°. 

From these measures it appears in a general way that the 
higher the latitude the greater the length of each astronomical 
degree. Thus, near the equator a degree has been found to be 
362800 feet in round numbers, while in Northern Sweden, in 
latitude 66°, it is 365800 feet. In other words, the earth's 
surface is flatter near the poles, as illustrated by Fig. 48. It is 
necessary to travel about 3000 feet farther in Sweden than 
in India to increase the latitude by one degree, as measured by 
the elevation of the celestial pole. 

Fig. 48. — Length of Degrees in Diiferent 

method of 
the earth's 
and form. 

Length of 
degrees of 
ical latitnde 
near tlie 



The following little table gives the length of a degree of the 
meridian in certain latitudes : 

At the equator one degree 

= 68.704 miles 

At lat. 20° 

= 68.786 • " 

u a 4QO u u 

1=68.993 " 

« u 450 _ a u 

= 69.054 " 

u u 60° " " 

= 69.230 " 

u u 80° " " 

= 69.386 " 

At the pole " " 

= 69.407 " 

It will be understood, of course, that the length of a degree 
at the pole is obtained by " extrapolation " from the measures 
made in lower latitudes. 

The difference between the equatorial and polar degree of 
latitude is more than 3500 feet, while the probable error of 
measurement cannot exceed more than three or four feet to 
the degree. 

134. The deduction of the exact form of the earth from 

Difficulty of such measurements is an abstruse problem. Owing to errors 

e ucmg ^£ observation, and to local deviations in the direction of 


results from gravity due to unevenness of surface and variation of den- 
sity in the rocks near the station, the different arcs do not 
give strictly accordant results, and the best that can be done 
is to find the result which most nearly satisfies all the obser- 

According to the determination of Colonel Clarke, for a long 
time at the head of the English Ordnance Survey, the dimen- 
sions of the " spheroid of 1866 " (which is still accepted by our 
Coast and Geodetic Survey as the basis of all its calculations) 
are as follows : 

the observa^ 

The earth's 
according to 

Equatorial radius 
Polar radius 


(A) 6 378206.4 meters 

(B) 6 356583.8 " 

21622.6 " 



= 3963.307 miles. 
= 3949.871 " 
= 13.436 " 

= 295.0 

THE p:arth as ax astroxomical body 119 

These numbers are likely to be in error as much, perhaps, as 
100 meters, and possibly somewhat more; the}^ can hardly 
be 300 meters wrong.^ 

A-B ^ , 

The oblateness — - — comes out 2 95 5 ^^^^ ^ comparatively 

small change in either the equatorial or polar radius would 
change the 295 by some units. 

At present the distance from a point on the earth's surface Uncertainty 
(say the observatory at Washington) to any point in the oppo- ^.^^^^^^^^^ 
site hemisphere (say the observatory at the Cape of Good Hope) 
is uncertain by fully 1000 feet. 

The deviation of the earth's form from a true sphere is due 
simply to its rotation, and might have been cited as proving it. 
As already shown, the centrifugal force caused by the rotation 
modifies the direction of gravity everywhere except at the 
equator and the poles, and the surface therefore necessarily 
takes the spheroidal form. 

135. Arcs of longitude are also available for determining the Availability 
earth's form and size. On an oblate or orange-shaped spheroid ?^^•?J^^| 
(since the surface lies wholly within the sphere which has or of any 
the same equator) the depTces of longitude are evidently every- ^^t®"^^^'® 

^ ^ ^ ^ _ . geodetic and 

where shorter than on the sphere, the difference being greatest astrouom- 
at a latitude of 45°, and from this difference when actually icai surveys. 
determined the oblateness can be computed. 

In fact, arcs in any direction between stations of ivJiicli both the 
latitude and longitude are known can be utilized for the purpose ; 
and thus all the extensive geodetic surveys^ that have been 

1 For Clarke's spheroid of 1878, see Appendix. 

2 It is extremely improbable that the actual geoid (the regular geometrical sur- 
face which most nearly fits the surface of the earth) is a perfect spheroid, or even a 
perfect ellipsoid of three axes. The local and continental irregularities are so great 
that it seems likely that it will be found best to adopt some one of the already 
computed spheroids as a final " surface of reference," and hereafter to investi- 
gate and tabulate the local deviations from this as a base, rather than to com- 
pute a new set of spheroid elements for every accession of new geodetic data. 



tion of form 
by pendn- 
lum obser- 

Loss of 
weight at 
equator is 
tIu as com- 
pared with 
weight at 
the pole. 


made by different countries contribute to our knowledge of 
the earth's dimensions. 

136. Determination of the Earth* s Form by Pendulum Experi- 
ments and purely Astronomical Observations. — Since ^, the time 

of vibration of a pendulum (Physics^ p. 79), equals 7r\_, we 
have 9 ^'^'^'W^ and can therefore measure the variations of g^ the 

force of gravity, at different parts of the earth by using a pendu- 
lum of invariable length and determining its time of vibration at 
each station. Extensive surveys of this sort have been made and 
are still in progress ; and it is found from them that the force of 
gravity at the pole exceeds that at the equator by about ■^-^. 
In other words, a man who weighs 190 pounds at the equator 
(weighed by a spring-balance) would weigh 191 at the pole. 

The centrifugal force of the earth's rotation accounts for about 
one pound in 289 of this difference ; the remainder (about one 
pound in bbb) has to be accounted for by the difference between 
the distances from the poles and from the equator to the center 
of the earth. At the pole a body is more than 13 miles nearer 
the center of the earth than at the equator, and as a conse- 
quence the earth's attraction upon it is greater. The difference 
of gravity between pole and equator depends, however, not 
only on the difference of distance from the center of the earth, 
but partly on the distribution of density within the globe. 

Assuming w^hat is probable, that tlie strata of equal density 
are practically concentric^ Clairaut has proved that for any planet 
of small oblateness, r^ _ ct\ ^ 

in which C equals the centrifugal force at the planet's equator 
and W the diminution of gravity between pole and equator; 
i.e., for the earth. 


21 X 

289 190 

which gives H 



The purely astronomical methods for determining the form of 
the earth indicate a slightly smaller oblateness of about -^^-q. Result of 
They depend upon precession and nutation (Sees. 168 and 170) P^^^'eiy^stio- 

'^ -^ . . ... . nomical 

and upon certain irregularities in the motion of the moon ; Ijut methods. 
their discussion lies quite beyond our scope. 

From a combination of all the available data of every kind, 
Harkness (1891) gives as his final result for the oblateness, 


n = 

300.2 ±3.0 

errors due 
to irreanlar 

137. Station Errors. — If the latitudes of all the stations in 
a triangulation as determined by astronomical observations are 
compared with their differences of latitude as deduced from station 
the geodetic operations, we find discrepancies by no means 
insensible. They are far beyond all possible errors of observa- distribution 
tions and are due to irregularities in the direction of gravity^ of matter in 

,.,, t 1 ... .,. ,„ p, crust of the 

which depend upon the variations m density and lorm oi the earth, 
crust of the earth in the neighborhood of the station. Such 
irregularities in the direction of gravity disjjlace the astronom- 
ical zenith of the station. They are called station errors and 
can be determined only by a comparison of astronomical posi- 
tions by means of geodetic operations. According to the Coast 
Survey, station errors average about 1''.5 in the eastern part 
of the United States, affecting both the longitudes and lati- 
tudes of the stations and the astronomical azimuths of the 
lines that join them. Station errors of from 4'' to 6'' are not 
very uncommon, and in mountainous countries these deviations 
occasionally rise to 30^' or 40''. 

138. Astronomical, Geographical, and Geocentric Latitudes. — Astronom- 
(1) The astronomical latitude of the station is that actually leai, geo- 
determined by astronomical observations, — simply the observed and geocen- 
altitude of the pole. trie latitudes 

(2) The geographical latitude is the astronomical latitude ,|istin- 
correeted for station error. It may be defined as the angle guished. 



latitude and 
the angle of 
the vertical. 

of geocentric 
degrees to 

formed with the plane of the equator by a hne drawn from the 
place perpendicular to the surface of the standard spheroid at 
that station. Its determination involves the adjustment and 
evening off of the discrepancies between the geodetic and astro- 
nomical results over extensive regions. The geographical 
latitudes (sometimes called topographical) are those used in con- 
structing an accurate map. 

For most purposes, however, the distinction between astronomical and 
geographical latitudes may be neglected, since on the scale of an ordinary 
map the station errors, amounting at most to a few hundred feet, would be 
entirely insensible. 

(3) Greocentric Latitude. While the astronomical latitude is 
the angle between the plane of the equator and the direction of 

gravity/ at any point, the geocentric lati- 
tude, as the name implies, is the angle, 
at the center of the earthy between the 
plane of the equator and a line drawn 
from the observer to that center; which 
line evidently does not coincide with 
the direction of gravity, since the earth 
is not spherical. 

In Fig. 49 the angle MNQ is the astro- 
nomical latitude of the point M (it is 
also the geographical latitude, provided the station error at that 
point is insensible), and MOQ is the geocentric latitude. 

The angle ZMZ\ the difference of the two latitudes, is called 
the angle of the vertical and is about llMn latitude 45°. 

Geocentric degrees are longest near the equator, in precise 
contradiction to the astronomical degrees ; and it is worth notic- 
ing that if we form a table like that of Sec. 133, giving the 
length of each degree of geographical latitude from the equator 
to the pole, the same table, read backwards., gives the length of 
the geocentric degrees without sensible inaccuracy; i.e.., at any 
distance from the pole a degree of geocentric latitude has within 

Fig. 49. — Astronomical and 
Geocentric Latitude 


a few inches just the same length as the astronomical degree at 
the same distance from the equator. 

Geocentric latitude is employed in certain astronomical cal- 
culations, especially such as relate to the moon and eclipses, in 
which it becomes necessary to "reduce observations to the 
center of the earth." 

139. Surface and Volume of the Earth. — The earth is so 
nearly spherical that we can compute its surface and volume (or 
" bulk ") with sufficient accuracy by the formulae for a perfect 
sphere, provided we put the earth's mean semidiameter for 
radius in the formulae. 

This mean semidiameter of an oblate spheroid is not — — , Meansemi- 

9 /y _[_ A diameter of 

but - — - — , because if we draw through the earth's center three an oblate 

o spheroid. 

axes of symmetry at right angles to each other, one only will 
be the axis of rotation, and both the others will be equatorial 

The mean radius r of the earth thus computed is 3958.83 
miles ; its surface (4 7rr^) is 196 944000 square miles, and its 
volume (I TTT^) 260000 million cubic miles, in round numbers. 


140. Definition of Mass. — The Mass of a body is the quantity Definition 
of matter in it, z.e., the number of tons^ pounds, or kilograms of ^ ™^^^' 
material which it contaiyis, — the unit of mass being a certain 
arbitrary body which has been selected as a standard. A '' kilo- 
gram," for instance, is the amount of matter contained in the 

block of platinum which is preserved as the standard at Paris. 
The mass of a body in the last analysis is measured by its its relation 
inertia, i.e., by the force required to give the body a certain ^^ force. 
velocity in a given time. 

Mass must not be confounded with " volume " or " bulk," 
which is simply the amount of space (cubic units) occupied by 



mass, on one 
hand, and 
volume and 
weight, on 
the other. 

arising from 
the identity 
of the ordi- 
nary names 
for the units 
of mass and 

the body. A bushel of coal has the same volume as a bushel 
of feathers, but its mass is immensely greater. 

Nor must mass be confounded with " weight," which is 
simply the force (push or pull) which urges the body towards 
the earth. It is true that under ordinary terrestrial conditions 
the mass of a body and its weight are proportional to each 
other and numerically equal. A mass of ten pounds iveighs 
(very nearly) ten pounds anywhere on the earth's surface, but 
the word " pound " in the two parts of the sentence means two 
entirely different things ; the pound of " mass " and the pound 
of " force " (stress) are as distinct as a " beam " of timber from 
a " beam " of sunlight. 

141. Mass and Force (Stress) distinguished. — This identity 
of names for the units of mass and force is on many accounts 
unfortunate, causing much ambiguity and much misunderstand- 
ing ; but its reason is obvious, because we usually measure 
masses by weighing^ and most often, not by weighing with a 
spring-balance, but by balancing the body against some standard 
mass, which standard is itself affected by variations of gravity 
in the same proportion as the body weighed, so that the ratio of 
their masses is correctly given notwithstanding such variations. 

The mass of a given body — the number of "mass units " in 
it — remains invariable, wherever it may be ; its weighty on the 
other hand — the number of "force units" which measures its 
tendency to fall, as judged by the effort required to lift it, or 
determined by a spring-balance — depends partly on where it 
is. At the equator it is nearly one half of one per cent less 
than at the pole, and on the surface of the moon it would be 
only about one sixth as great as on the earth. 




To use an illustration suggested by Professor Newcomb : Suppose a 
base-ball team could somehow get to the moon. They would find their 
bats and balls very light to lift and hold ; they themselves would be light 
on their feet and could jump six times as high and as far as on the earth, 
gravity and iceight being so much less than here. But, since masses remain 


unchanged, the pitcher could not with a given exertion send the ball any 
more swiftly than here, nor could the batter hit it any harder or give it 
greater speed (though it would fly much farther before it fell), and the 
catcher in capturing the ball would receive just the same blow upon his 
hands as here. And if they had a steak for dinner that " weighed " only 
two pounds on their spring-balance, it would give them twelve pounds 
of meat ; and, we may add, would also " weigh " twelve pounds on a 
platfoi^m scale, or steelyard. 

The student must always be on his guard whenever he comes 
to the word "pound" or "kilogram," or any of their congeners, 
and must consider whether he is dealing with units of mass or 
of force. 

142. The Scientific Unit of Force or Stress, — the Dyne, Mega- 
dyne, and Poundal. — Many high authorities now urge the entire 
abandonment of the old force units which bear the same names 
as the mass units, and the substitution, in all scientific work at 
least, of the dyne (Physics^ p. 18) and its derivative, the mega- 
dyne. The change would certainly conduce to clearness, but 
for a time at least would be inconvenient, as making former 
mechanical literature almost unintelligible to those familiar 
with the new units only. 

The Dyne is the force (pull or push) which acting constantly The scien- 
foT one second upon a mass of one gram would give it a velocity ^^^^ ^^^* ^^ 

T-r ii'i?i-r»- force: the 

of one centimeter a second, it equals the " weight (at raris) ^yne. 
of a mass of , or 1.0199, milligrams. The Megadyne (a 

million dynes) is the weight (at Paris) of a mass of 1.0199 
kilograms, or almost exactly 1.02 kilograms in the latitude of 

Many English authorities, however, insist on a unit of force 
based on the British units of mass and length and employ 
the Poundal^ — the force which in one second would give a The English 
velocity of 07ie foot per second to a mass of one pound. Since P^^^^^^- 

1 9.805 meters per second is the value of g at Paris. 



The law of 
in words. 

The mystery 
of attraction 
not yet 

attract each 
other as if 
their masses 
were all 
at their 

g equals (nearly) 32^ feet per second, the poundal is about 
-— - of the " weight " of a mass pound at London, or very 

nearly half an ounce of " pull." More accurately, the poundal 
equals 13.865 dynes. 

143. Gravitation. The Cause of Weight. — Science cannot 
yet explain why bodies tend to fall towards the earth and push or 
pull towards it when held from moving ; but Newton discovered 
and proved that the phenomenon is only a special case of the 
much more general fact which he inferred from the motions of 
the heavenly bodies and formulated as the Law of G-ravitation^ 
under the statement that any two particles of matter attract 
each other with a force proportional to their masses and inversely 
proportional to the square of the distance between them. 

If instead of particles we have bodies composed of many 
particles, then the total force between the bodies is the sum of 
the attractions of all the different particles, each particle attract- 
ing every particle in the other body and being attracted by it. 

We must not imagine the word attract to mean too much. It 
merely states as a fact that there is a tendency for bodies to move 
towards each other, without including or implying any explana- 
tion of the fact. Thus far none has appeared which is less diffi- 
cult to comprehend than the thing itself. It remains at present 
simply a fundamental fact, though it is not impossible (nor im- 
probable) that ultimately it shown to be a necessary con- 
sequence of the relation between particles of ordinary matter and 
the all-pervading "ether" to which we refer the phenomena of 
light, radiant heat, electricit};^, and magnetism {Physics^ p. 315). 

144. The Attraction of Spheres. — If the two attracting bodies 
are spheres, either homogeneous or made up of concentric shells 
which are of equal density and thickness throughout, then, as 
Newton demonstrated, the action on bodies outside the sphere 
is precisely the same as if all the matter of each sphere were 
collected at its center. If the distance between the bodies is very 


great compared with their size, then, whatever their form, the 
same thing is nearly, though not exactly, true. 

He also showed that within a homogeneous hollow sphere of Attraction 
equal density and thickness throughout the attraction is every- ^®^° withm 
where zero; i.e., a body anywhere within the hollow shell would sphere. 
not tend to fall in any direction. 

If bodies which attract each other are prevented from moving, 
the effect of the attraction will be a stress (a push or pull), to be 
measured in dynes or force units (not in mass units), and this 
stress is given by the equation which embodies the law of gravi- Law of 

tation, viz., gravitation 

_ M\ X M^ expressed as 

F (dynes) = Q y^ -^ > an equation 

giving the 

where M^ and M^ are the masses of the two bodies (expressed ^s TpuiT 
in grams), d is the distance between their centers (in centi- in dynes. 
meters), and (^ is a factor known as the Newtonian Constant or 
the Constant of Gravitation. 

145. The Constant of Gravitation. — The constant of gravi- Thecon- 
tation is believed to be, like the velocity of light, an absolute stantof 
constant of nature, — the same in all the universe of matter, — fhelame^^ 
among the stars and planets as well as upon the earth. This is not everywhere 
yet absolutely proved, but there is no known phenomenon that !!i^,,l«. 
contradicts it, and there is much probable evidence in its favor. 

The numerical value of G depends on the units of mass, 
distance, and time; and in the C.G.S. system (centimeter- 
gram-second) it is, according to the elaborate determination 
of Boys in 1894, 0.00000006657 (6657 x 10"") or, quite 
within the limits of experimental error, one fifteen-millionth, its numeri- 
If, for instance, the mass M. is 1000 grams, M^ 2000 grams, ^ai value 

OTiQ liftcan- 

and the distance 10 centimeters, the force in dynes will be millionth 

in the C.G.S. 
1 ^1000X2000^.^ 1 ^^^^^^^ systen^of 

15 000000 100 15 000000 ' ^^^ts 
or yi^ of a dyne. 



giving the 
force due to 
in centi- 
meters per 

It may be added that it is not yet proved that the equation F = G x M^ 
X M^ -f- d'^ is the complete law. It is conceivable (though highly improb- 
able) that the right-hand member may be only the principal term of a 
series which contains other terms (involving temperature, for instance) 
that may become sensible under conditions widely different from any yet 
observed. And " matter " may exist which does not " gravitate " though 
possessing "inertia," — the ether, for instance. 

146. Acceleration by Gravitation. — If Jfj and M^ are set free 
while under each other's attraction, they will at once begin to 

approach each other. At the end of the first second M^ will 

have acquired a velocity F^ = (^ X --|, which, the student will 

observe, depends entirely upon the mass of M^ and not at all 
upon that of M^ itself. (A grain of sand and a heavy rock 
will fall at the same rate in free space under the attraction of a 
given body when at the same distance from it.) 


Similarly, M^ will have acquired a velocity Fg = 6^ X --^. 

The speed with which the bodies approach each other will be the 
sum of these velocities, and if we denote this relative accelera- 
tion (or the velocity of approach acquired in one second) by/, 
we shall have 

M^ + M^ 

d' ) 

This is the form of the law of gravitation which is used in 
dealing with the motions of the heavenly bodies, caused by their 

Notice that while the expression for F (the stress in dynes) 
has the product of the masses in its numerator, that for / (the 
relative acceleration) has their sum^ and, like g^ is expressed in 
centimeters per second. 

147. We are now prepared to discuss the methods of meas- 
uring the earth's mass. It is only necessary to compare the 
attraction which the earth exerts on a body, m, on its surface 
(at a distance, therefore, of 3959 miles from its center) with the 


attraction exerted upon m hy some other body of a known mass Mass of 

at a known distance. The practical difficulty is that the attrac- ^^^'^^ meas- 
ured by com- 
tion of any manageable body is so extremely small, compared paring its 

with the attraction of the earth, that the experiments are attraction 

exceedingly delicate. Unless the mass employed for compari- ,^^ j^.^ g^j.. 

son with the earth is one of several tons, its attraction will be face with 

only a fraction of a dyne, — hard to detect even, and worse to i^no^^ ^^^^ 

measure. attracting 

The various experimental methods which have been actually \ ^^^me 

^ -^ body at a 

used thus far for determining the earth's mass are enumerated known 
and discussed in the author's General Astronomy^ to which the distance. 
student is referred. We present here only a single one, which 
is not difficult to understand and is probably the best, though 
not quite the earliest. 

148. The Earth's Mass and Density determined by the Torsion The torsion 
Balance. — This is an apparatus invented by Michell and first ^^i^^<^^- 
employed by Lord Cavendish in 1798. A light rod carrying 
two small balls at its extremities is suspended at its center by 
a fine wire, so that the rod will hang horizontally. If it be 
allowed to come to rest, and then a very slight deflecting force 
be applied, the rod will be drawn out of position by an amount 
depending on the stiffness and leiigth of the wire as well as the 
intensity of the force. When the deflecting- force is removed 
the rod will vibrate back and forth until brought to rest by the 
resistance of the air. 

The Coefficient of Torsion^ as it is called, — z.e., the stress Deten 


which will produce a twist of one revolution, — can be accu- tionofthe 
rately determined by observing the time of free vibration when of torsion 
the dimensions and masses of the rod and balls are known (see ^J observa- 
Anthony and Brackett, Physics^ p. 117), and this coefficient will vibration, 
enable us to determine what force in dynes is necessary to pro- 
duce a twist or deflection of any number of degrees. If the 
wire is stiff the coefficient will be large, and correspondingly 
small if very slender. It is therefore desirable that it should 



of deflection 
caused by 

of disturb- 
ing causes. 

a' a a' 

o o 

be as slender as possible, while yet sufficiently strong to sustain 
the rod and balls. 

149. The Observations. — If, now, after the torsional coeffi- 
cient has been carefully determined by observing the free vibra- 
Observation tions of the rod, two large balls, A and B (Fig. 50), are brought 
near the smaller ones, a deflection will be produced by their 
attraction, and the small balls will move from a and h (their 

position of rest) to a^ and h'. By 
shifting the large balls to the other 
side at A' and B' (which can be 
done by turning the frame upon 
which they are supported) Ave shall 
get a similar deflection in the oppo- 
site direction, — z.e., from a^ and h' 
to a" and h'\ — and the difference 
between the two positions assumed 
by the two small balls — i.e.^ the 
distance a' a" and h'h'^ — will be 
twice the deflection which is due 
to the attraction of the two large 
balls for the two small balls. 

The observations of vibration 

and deflection are best made by 

watching with a telescope from a 

distance the reflection of a fixed scale in a little mirror attached 

to the horizontal beam at C. 

In observing the deflections it is not necessary, nor even best, 
to wait for the balls to come to rest. While still vibrating 
we note the extremities of their swing. The middle point of 
the swing (with a slight correction) gives the place of equilib- 
rium. We must also carefully determine the distances Aa\ 
A'V\ Bb', and B'a" between the center of each of the large balls 
and the center of the small ball when deflected. The utmost 
care must be taken to exclude air currents and electrical or 

Fig. 50. — Plan of the Torsion 



mag-netic disturbances, since these would seriously vitiate the 

150. Calculation of the Earth^s Mass from the Experiment. — 
The earth's attraction on each of the small balls, m, is evidently 
measured by the ball's weighty w, corrected for the centrifugal 
force of the earth's rotation at the observer's station and 
reduced to dynes. 

The attractive force of the large ball on the small one near it Calculation 
is found from the observed deflection. If, for instance, this ^^^^^^h's 
deflection is one degree, and the coefficient of torsion is such 
that it takes a force of ten dynes at the end of the rod to twist 
the wire one whole turn, then the deflecting force, which we 
will call 2/, will be Jg of a dyne. One half of this deflecting 
force, /, will be due to ^'s attraction of <x, and half to i^'s 
attraction of h. Call the mass of the large ball 7?, determined 
by its weight, and that of the small ball m, and let d be the 
measured distance Bh' between their centers. We shall then 
have, from Sec. 144, the equation 

Similarly, calling E the mass of the earth and B its radius, w 
being the corrected weight in dynes of the small ball, we shall 

w=G[ -— ), or^ = (h) 

\ B^ J G-m ^ ^ 

Dividing equation (a) by (6), G and m cancel out, and we have 

which gives the mass of the earth in terms of B. 

By the same observations the value of the constant of gravi- Determina- 

f , ^2 tion of con- 

tation is determined, since, from equation (a), G = "^ , B and stantof 

^ ' '^^ gravitation. 
m being both measured in grams, and d in centimeters. 



151. Density of the Earth. — Having the mass of the earth 
it is easy to find its density. The volume, or bulk, in cubic 
miles has already been given (Sec. 139) and can be reduced to 
cubic feet by simply multiplying that number by the cube of 
5280. Since a cubic foot of water contains 62^ mass pounds 
(nearly), the mass which the earth would have, if composed of 
water, follows. Comparing this with its mass obtained from the 
observations, we get its density. 

A combination of the results of all the different methods 
hitherto employed, taking into account their relative accuracy, 
gives about 5.53 as the most probable value of the earth's 
density compared with water, but the second decimal is not 

Direct 152. Density determined directly. — We can deduce the earth's density 

determina- directly from the observations without any preliminary calculation of its 
^^^^ °^ mass. 

jx ^rithout Letting r and 6^ represent the radius and density of the ball B, its mass 
previous is ^irr^s. Similarly, E, the mass of the earth, is f TritV, / being the 

calculation earth's mean density. 

of its mass. Substituting these values of B and E in equations (a) and (h) of the 

preceding section, we have 

..3, - f!t^ 


f 7rr-5 

^irRh' = 


Dividing (<i) by (c), we get 
whence, finally, 

72V w7?2 



w r r^ 
= sx-x — X— , 

/ R d'^ 

Early experi- 

giving the density of the earth in terms of the density of the ball B, and 
other known quantities. 

153. In the earlier experiments, by this torsion-balance method, the 
small balls were of lead 1 or 2 inches in diameter, at the extremities of 
a light wooden rod 5 or 6 feet long inclosed in a case with glass ends, and 
their positions and vibrations were observed by a telescope looking directly 
at them from a distance of several feet. The attracting masses, A and B^ 


were balls (also of lead) about a foot in diameter. Great difficulties were 
encountered from currents of air within the inclosure. 

Boys in 1894 used a most refined apparatus in which the small balls (of Experi- 
gold), one quarter of an inch in diameter, were hung at the end of a beam i^^ents of 
only a centimeter long, which was suspended by a delicate fiber of amor- ^° ®^^^^ 
jyJwus quartz, an ingenious invention of the experimenter. The deflections 
due to the attraction of two sets of lead balls, respectively 2^ and 4^ inches 
in diameter, were measured by observing with a telescope the reflection of 
a scale in a little mirror attached to the beam. The whole apparatus was 
placed in an air-tight case no larger than an ordinary water pail, from 
which the air was exhausted and a little hydrogen admitted to take its 
place. His result for the earth's density was 5.527. It was from these 
experiments that the value of the constant of gravitation already given was 

Different values for the earth's density, obtained by experiments during 
the last fifty years, range all the way from 5.8 to 4.9, omitting one or two 
which are very discordant from circumstances easily understood. 

154. Constitution of the Earth's Interior. — Since the average 
density of the earth's crust does not exceed three times that of 
water, while the mean density of the whole earth is about 5.5, increase 
it is obvious that at the earth's center the density must be very °^ density 

^ ^ '^ towards 

much greater than at the surface, — very likely as high as eight earth's 
or ten times the density of water, and perhaps higher, — equal center. 
to that of the heavier metals. There is nothing surprising in 
this. If the earth were ever fluid, it is natural to suppose 
that before solidification took place the densest materials would 
settle towards the center. 

Whether the interior of the earth is solid or fluid, it is difli- 
cult to say with certainty. Certain tidal phenomena, to be 
mentioned hereafter, have led Lord Kelvin to express the 
opinion that " the earth as a whole is solid throughout, and Question 
more rigid than glass," volcanic centers being mere " pustules," ^^ to the 
so to speak, in the general mass. To this most geologists plasticity of 
demur, maintaining that at a depth of not many hundred miles ^^® central 
the materials of the earth must be fluid or at least semi-fluid. 
This is inferred from the phenomena of volcanoes and from the 


fact that the temperature continually increases with the depth 
so far as we have yet been able to penetrate. But thus far the 
deepest penetration is but little more than a single mile, — a 
mere scratch, — not 3- 5^o"o P^^'^ ^^ ^^® distance to the center of 
the earth. 


1. Does the transportation of sediment by the Mississippi tend to 
lengthen or to shorten the day? 

2. If the diameter of the earth were doubled, keeping its mass 
unchanged, how would its density and the weight of bodies at its surface 
be affected? 

3. If its diameter were trebled, keeping its density unchanged, how 
much would its mass and the weight of bodies at its surface be increased? 

4. Supposing the earth to be homogeneous, how great (approximately) 
would be the force of gravity 1000 miles below its surface? 

Solution. Inside of a hollow sphere the attraction is zero (Sec. 144). At 
the depth of 1000 miles, therefore, the effective attraction would be that 
of a sphere of only 3000 miles radius, all the shell of matter outside of this 
being without influence. We should therefore have gravity at the surface : 

gravity 1000 miles below the surface z= f !^ : 4 ^ (^ - IQOO)^ = R:{R- 1000) ; 

' i?2 ^ (E - 1000)'-^ 

i.e., as 4000 :3000. In words, the attraction at this depth would be | that at 

the surface of the earth, if it were of equal density throughout, which it is not. 

5. Assuming g at the earth's surface to be 9.805 meters per second, 
what would it be in a balloon at an elevation of 2 miles ? The radius of 
the earth may be taken as 4000 miles and centrifugal force neglected. 

Ans. 9.7952 m per second. 
Would the value of g be the same at the top of a mountain 2 miles 
high, and if not, why? 

6. Given two spheres one of which has a mass m times greater than the 
other ; on what point on the line joining their centers are their attractions 

Solution. Let d be the distance between their centers and x the distance 
of the point of equilibrium from the smaller body ; then the attraction of the 

171 1 

larger body at that point is G ■ , that of the smaller being G — • Cancel- 

{d — x)2 /— x^ 

ing the (x's and taking the square roots, we have =: -■> from which we 

have (^-^) ^ 

Ans. X = -=. 

1 + V m 



7. Assuming the moon's mass as g\ of the earth's, where is the equi- 
librium point on the line of centers? 

Ans. At a point j\ of the distance from the moon to the earth. 

8. What is the attraction in dynes between two spheres, each having a 
mass of 1000 kilograms, at a distance of 1 meter between centers? 

Ans. 6f dynes, or the weight of 6.8 mgm. 

9. If these spheres were free to move under their mutual attraction, 
required their relative velocity at the end of one second. 

Ans. j-^QQ mm per second. 

10. If at a distance of half a meter from such a ball is suspended a 
small ball weighing 1 gram, what is the attraction between them ? 

Ans. fyioo of a dyne. 

11. If in this case the small ball were suspended by a fine thread 
10 meters long, how many millimeters would it be drawn from a vertical 
position, and what angle would the thread make with the vertical? 

Deviation, 0.000272 mm. 


Deflection, 0''.00561. 

Yerkes Observatory 



Sun's appar- 
ent motion 
in declina- 

Its motion 
in right 

The Apparent Motion of the Sun, and the Orbital Motion of the Earth — Precession 
and Nutation — Aberration — The Equation of Time — The Seasons and the 

155. The Sun's Apparent Annual Motion among the Stars. — 

This must have been among the earliest recognized of astronom- 
ical phenomena, as it is obviously one of tlie most important. 

As seen by us in the northern hemisphere, the sun, starting 
in the spring at the vernal equinox, mounts higher in the sky 
each day at noon for three months, until the summer solstice, 
and then descends towards the south, reaching in the autumn 
the same noonday elevation which it had in the spring. It 
keeps on its southward course to the winter solstice in Decem- 
ber and then returns to its original height at the end of a year, 
marking and causing the seasons by its course. 

Nor is this all. The sun's motion is not merely north and 
south, but it also advances continually eastward among the 
stars. In the spring the stars which at sunset are rising on the 
eastern horizon are different from those which are found there 
in summer or winter. 

In March the most conspicuous of the eastern constellations at sunset 
are Leo and Bootes. A little later Virgo appears, in the summer Ophiu- 
chus and Libra ; still later Scorpio, while in midwinter Orion and Taurus 
are ascending as the sun goes down. 

So far as the obvious appearances are concerned, it is quite 
indifferent whether we suppose the earth to revolve around the 
sun or vice versa. That the earth is the body which really 
moves, however, is absolutely demonstrated by two phenomena 



too delicate for observation without the telescope, but accessible Facts which 
to modern methods. The more conspicuous of them is the *^^^"o^|st^'^^^ 

^ that the 

aberration of fight; the other is the annual parallax of the fixed apparent 
stars. These phenomena can be explained only by the actual ^^^^'^^^ ^^ 
motion of the earth and will be discussed later. clue to the 

156. The Ecliptic; its Related Points and Circles. — By observ- leai motion 

T .-, •11 • T •1111 111- • of the earth. 

mg daily with the meridian-circle both the sun s decimation 

and also the difference between its right ascension and that of 

some chosen star, we obtain a series of positions of the sun's 

center which can be plotted on a globe, and we can thus mark 

out its path among the stars. It is a great circle, called the 

Ecliptic (Sec. 23), which cuts the equator at two opposite The ecliptic 

points (equinoxes), at an angle of approximately 23^° (23° 27^ ^^ "^*^' 

8''.0 in 1900). It gets its name from the fact, early discovered, 

that eclipses happen only when the moon is crossing it. It is 

the great circle in which the plane of the earth^s orbit cuts the 

celestial sphere. 

The angle which the ecliptic makes with the equator at the 
equinoctial points is called the Obliquity of the Ecliptic and is Obliquity of 
evidently equal to the suns maximum declination^ reached in *^^® ecliptic. 
June and December. 

The two points in the ecliptic midway between the equinoxes 
are called the Solstices, because there the sun apparently Solstices 
" stands," i.e., stops and reverses its motion in declination. ^"^^ tropics. 
The circles drawn through these solstices parallel to the 
equator are called the Tropics. 

The Poles of the Ecliptic are the two points in the heavens The poles of 
90° distant from every point in that circle. The northern one ^^^ ^^^^pt^^'- 
is the constellation Draco, about midway between the stars 
8 and f, and on the solstitial colure (right ascension, 18 hours), 
its distance from the pole of rotation being equal to the obliquity 
of the ecliptic (see Sec. 27). 

It will be remembered that celestial latitude and longitude are 
measured with reference to the ecliptic and not to the equator. 



Th,e zodiac. 

Signs of the 

between tlie 
ecliptic and 
tlie orbit of 
the earth. 

tions for 
the form of 
tlie orbit. 

157. The Zodiac and its Signs. — A belt 16° wide (8° on each 
side of the ecliptic) is called the zodiac^ or " zone of animals " 
(German, Thierkreis), the constellations in it, excepting Libra, 
being all figures of living creatures. It is taken of that par- 
ticular width simply because the moon and the principal planets 
always keep within it. It is divided into the so-called " signs," 
each 30° in length, having the following names and symbols : 

' Aries ^ 

Spring I Taurus y 

Gemini n 

f Cancer ee 
Summer i Leo ^ 

[ Virgo TT)^ 

' Libra 
Autumn i Scorpio 




[ Sagittarius ^ 

' Capricornus >J 

Aquarius x^ 

Pisces X 

The symbols are for the mpst part conventional pictures of the objects. 
The symbol for Aquarius is tjie Egyptian character for water. The origin 
of the signs for Leo, Capricornus, and Virgo is not quite clear. 

The zodiac is of extreme antiquity. In the zodiacs of the 
earliest history the Lion, Bull, Ram, and Scorpion appear 
precisely as now. 

158. The earth^s orbit is the path in space pursued by the 
earth in its revolution around the sun. The ecliptic is not the 
orbit and must not be confounded with it ; it is simply a great 
circle of the infinite celestial sphere, while the orbit itself is 
(nearly) a circle, of finite diameter, in space. The fact that the 
ecliptic is a great circle gives us no information about the orbit, 
except that it lies wholly in one plane, which passes through the 
sun; it tells us nothing as to the orbit's xeslform or size. 

By reducing the daily observations of the sun's right ascen- 
sion and declination made with a meridian-circle to celestial 
longitude and latitude (the latitude would always be exactly 
zero, except for some slight perturbations of the earth) and 
combining these data with observations of the sun^s apparent 
diameter, we can, however, ascertain the form of the earth's 



orbit and the law of its motion in this orbit. The size of the 
orbit cannot be fixed until we find some means of determining 
the scale of miles. 

159. To find the Form of the Orbit Take a point, S (Fig. 51), 

for the sun, and draw from it a line, SO^ directed towards the 
vernal equinox, from which longitudes are measured. Lay 
off from S lines indefinite in length, making angles with SO 
equal to the earth's longitude as seen from the sun ^ on each of 
the days when observations were made. We shall thus get a 
sort of " spider," showing 
the direction of tlie earth as 
seen from the sun on each 
of those days. 

Next, as to the distances. 
While the apparent diameter 
of the sun does not deter- 
mine its absolute distance 
from the earth unless we 
know the diameter in miles, 
yet the changes in the appar- 
ent diameter do inform us 
as to the relative distance 
at different times, — the distance being inversely proportional 
to the sun's apparent diameter (Sec. 10). If, then, on this 
"spider" we lay off distances equal to the quotient obtained 
by dividing some constant, say 10000, by the sun's apparent 
diameter in seconds as observed at each date, these distances 
will be proportional to the true distance of the sun from the 
earth, and the curve joining the points thus obtained will he a 
true map of the eartVs orbit, though without any scale of miles. 

When the operation is performed, we find that the orbit is 
an ellipse of small eccentricity (about g^), with the sun not in 
the center, but at one of the two foci. 

1 This is 180° ± the sun's longitude as seen from the earth. 

Direction of 
earth from 
sun on days 
of observa- 

Fig. 51. 

■ Determination of the Form of the 
Earth's Orbit 

from sun on 
these days. 



Definition of 
the ellipse, 
its axes, and 

Definition of 




vector, and 


of the eccen- 
tricity of 
the earth's 
orbit by 

160. Definitions relating to the Orbital Ellipse. — The Ellipse 
is a curve such that the sum of the two distances from any point 07i 
its circumference to two jjoi^its within, called the foci, is always 
constafit and equal to the major axis of the ellipse. 

In Fig. 52, wherever P is taken on the periphery of the 

elhpse, tSF + FF always equals AA\ which is the major axis. 

AC is the semi-major Axis and is usually denoted by A or a. 

BC is the semi-minor Axis, denoted by ^ or h; the eccentricity 

of the ellipse is the fraction, or ratio, -^^, and is usually 

expressed as a decimal. It equals 


"V ^2 _ ^2 


Fig. 52. — The Ellipse 

If a cone is cut across obliquely by a plane, the section is an 

ellipse, which is therefore called 
one of the conic sections. (See 
Sec. 314.) 

Perihelioyi and Aphelion are 
respectively the points where 
the earth is nearest to and 
remotest from the sun, the line 
joining them being the major 
axis of the orbit. The Line of 
Apsides is the major axis indefi- 
nitely produced in both directions. A line drawn from the 
sun to the earth or any other planet at any point in its 
orbit, as SP in the figure, is called the planet's Radius Vector, 
and the angle ASF, reckoned from the perihelion point, in 
the direction of the planet's motion towards B, is called its 

161. Discovery of the Eccentricity of the Earth's Orbit by 
Hipparchus. — The variations in the sun's diameter are too 
small to be detected without a telescope, so that the ancients 
failed to perceive them. Hipparchus, however, about 120 B.C., 
discovered that the earth is not in the center of the circular 



orbit,^ which he supposed the sun to describe around it with 
uniform velocity. 

Obviously, the sun's apparent motion is not uniform, because 
it takes 186 days for the sun to pass from the vernal equinox 
to the autumnal, and only 179 days to return. Hipparchus 
explained this difference by the hypothesis that the earth is out 
of the center of the circle. 

In fact, the earth's orbit is so nearly circular that the differ- 
ence between the radius vector of the ellipse and that of the 
eccentric circle of Hipparchus is everywhere so small that the 
method indicated in the preceding article would not practically 
suffice to discriminate between them. Other planetary orbits 
are, however, unmistakable ellipses, 
and the investigations of Newton 
show that the earth's orbit also is 
necessarily elliptical. 

162. The Law of the Earth»s 
Motion. — By comparing the meas- 
ured apparent diameter of the sun 
with the differences of longitude 
from day to day, we can deduce 
not only the form of the orbit, but 
the law of the eartlis motion in it. 
motions and apparent diameters 

Fig. 53. 

Equable Description 
of Areas 


On arranging the daily 
a table, we find that 
the daily motions vary directly as the squares of the diame- 
ters^ or inversely as the squares of the distajices of the earth 

The law of 
the earth's 

from the sun. In other words, the product of the square of motion — 

the law of 
equal areas 

the distance hy the daily motion is constant. Now the area 
of any small elliptical sector cSd (Fig. 53) which is sensibly 
a triangle = )y Sc • Sd sin cSd, or l- r'r" sin cSd. When the 

in equal 

1 Until the time of Kepler, it was universally assumed on metaphysical 
grounds that the orbits of the celestial bodies must necessarily be circular and 
described with a uniform motion, "because," as was reasoned, "the circle is 
the only perfect curve, and uniform motion is the only perfect motion proper to 
heavenly hodies.'''' 



tion of the 
law of areas 
from obser- 

Proved by 
Newton to 
be a neces- 
sary conse- 
quence of 


angle is small r' x r" = (sensibly) r^, r being the " radins vector " 
drawn to the middle of cd ; and for sin cSd we may put eSd 
itself. Hence, area cSd = l r^ x cSd^ — a constant, as obser- 
vation shows; or, in other words, its radius vector describes 
areas proportional to the times, a law which Kepler first discov- 
ered in 1609. 

If in Fig. 53 ah, cd, and ef be portions of the orbit described 
by the earth in different weeks, the areas of the elliptical sectors 
aSh, cSd, and eSf are all equal. A planet near perihelion 
moves faster than at aphelion in just such proportion as to pre- 
serve this relation. 

163. As Kepler left the matter this is a mere fact of obser- 
vation. He could give no reason for it. Newton afterwards 

proved that it is a necessary con- 
sequence of the fact that the earth 
moves under the action of a force 
always directed towards the sun 
(Sees. 303 and 304). The law 
holds good in every case of orbital 
motion under a central attraction 
and enables us to find the position 
of the earth or any planet, at any 
time, when we once know the time of its orbital revolution, or 
"period," and a date when it was at perihelion. Thus, the angle 
ASF (Fig. 54), or the anomaly of the planet, must be such that the 
shaded area of the elliptical sector ASF will be that portion of 

the whole ellipse which is represented by the fraction — , T being 

the number of days in the period and t the number of days 
since the planet last passed perihelion. The solution of this 
problem, known as Kepler's Problem, leads to a " transcen- 
dental " equation, and can be found in all books on phys- 
ical astronomy, or in the Appendix to the author's General 

Fig. 54. — Kepler's Problem 


164. Changes in the Orbit of the Earth. — Were it not for the 
attraction of the planets upon the earth and sun, the earth would Effect of 
maintain her orbit strictly unaltered from as^e to ag-e, except ^ ^"etary 

•^ . ° ° ^ attractions 

that possibly in the course of millions of years the effect of upon the 

falling meteors and the attraction of some of the nearer stars ^^^^^^'^ 
might become barely sensible. In consequence, however, of 

the attractions of the other planets, it is found that, while the Major axis 

Major Axis of the orbit and the Length of the Year remain in ^"f^pei'iod 

•^ ^ -^ ^ ^ unaffected 

the long run unchanged^ other elements undergo slow variations in the 
known as " secular perturbations." ^^^^ ^'^'^• 

(1) Revolution of the Line of Apsides. This line, which now 
stretches in both directions towards the opposite constellations 

of Gemini and Sagittarius, moves continually eastward (z.e., in Eastward 
the same direction as the planetary motions) at a rate which ^'evolution 

. . of the line 

would carry it completely around the circle in about one of apsides. 
hundred and eight thousand years, if the rate continued 
always the same as at present, — which, however, it will not, 
since it is affected by changes in the eccentricity and by other 

(2) Change of Eccentricity. At present the eccentricity of Change of 
the earth's orbit, now 0.016, is slowly diminishing", and accord- eccentricity 

*^ ^ — at present 

ing to Leverrier will continue to do so for about twenty-four slowly 
thousand years, when it will be only 0.003; z.e., the orbit will diminishing. 
become almost circular. Then it will increase again for some 
forty thousand years and will continue to oscillate, always keep- 
i7ig between zero and 0.07. But the successive oscillations are 
not equal either in amount or time, — not at all like the " swing 
of a mighty pendulum," which has been rather a favorite figure 
of speech with some writers. 

(3) Change in the Obliquity of the Ecliptic. The plane of Obliquity of 
the earth's orbit is also slowly changing its position, and as a *j^® echptic 
consequence the ecliptic shifts its place among the stars, thus diminishing, 
slowly altering their latitudes and the angle between the equator 

and the ecliptic. The obliquity is now about 24' less than it 


was two thousand years ago,^ and at present is decreasing about 

0".6 yearly. After about fifteen thousand years, when the 

obliquity will be only 22^°, it will begin to increase again and 

will " oscillate " like the eccentricity. But the whole change 

can never exceed about l2° on each side of the mean. 

Slight peri- (4) Periodic Disturbances of the Earth in its Orbit. Besides 

odic disturb- \\-^q^q "sccular perturbations" of the earth's orbit^ the earth 

earth in itsclf IS all the time slightly disturbed m its orbit. On account 

its orbit. of its connection with the moon, its center oscillates each month 

a few hundred miles above and below the true plane of the 

ecliptic ; and by the action of the other planets it is sometimes 

set forward or backward or sideways to the extent of several 

thousand miles. Of course, every such displacement of the 

earth produces a corresponding slight change in the apparent 

position of the sun, and indeed of all bodies observed from the 

earth, except the moon, which accompanies the earth, and the 

stars, which are too far away to be sensibly affected. 

165. Precession of the Equinoxes. — This is a slow ivestward 
Precession motion of the equinoxes along the ecliptic first discovered by Hip- 
defined. Its parchus about 125 B.C. He found that the "year of the seasons," 
Hipparchus. froHi solsticc to solstice, as determined by the gnomon, was shorter 
than that determined by the heliacal rising and setting of the stars 
{i.e., the times when certain constellations rise and set at sunset), 
just as if the equinox "preceded," i.e., "stepped forward," a 
Amount of little to meet the sun. The difference between the year of the 
precession scasons and the sidereal year is about twenty minutes, the preces- 
Period 25800 ^*^^ being 50''. 2 yearly, so that the equinox makes the complete 
years. circuit of the ecliptic in twenty-five thousand eight hundred years. 

It is a motion of the equator and not of the ecliptic which 

1 The ancients determined the "obliquity" with fair accuracy by observa- 
tions of the length of the shadow of the gnomon at the two solstices. The 
angle CBB, or SBS' (Sec. 93, Fig. 36), is twice the obliquity. The Chinese 
records contain an observation which purports to be four thousand years old 
and is apparently genuine. 



causes the precession, as is proved by the fact that the latitudes 
of the stars have changed but slightly in the last two thousand 
years, showing that the ecliptic maintains its position among 
them nearly unaltered. The right ascension and declination of 
the stars, on the other hand, are both found to be constantly 
changing, and this makes it certain that it is the celestial equator 
which shifts its place. On account of this change in the place 
of the equinox the longitudes of the stars increase continually, 
— at a sensibly constant 
rate of 50". 2 a year, — 
nearly 30° in the last 
two thousand years. 

166. Motion of the Pole 
of the Heavens around 
the Pole of the Ecliptic. — 
The obliquity of the eclip- 
tic^ which equals the an- 
gular distance of the pole 
of the heavens from the 
pole of the ecliptic, is 
not affected by preces- 
sion. That is to say, as 
the earth travels around 
its orbit in the plane of 
the ecliptic (just as if 
that plane were the level surface of a sheet of water in which 
the earth swims half immersed), its axis, ACX (Fig. ^b), always 
preserves very nearly the same constant angle of 23i° with the 
perpendicular, SCT, which points to the pole of the ecliptic. 
But, in consequence of precession, the axis, while keeping its 
inclination unchanged, shifts conically around the line SCT 
(like the axis of a spinning top), taking up successively the 
positions AK\ A''C, etc., thus carrying the equinox from Fto ^^ 
and onwards. 

Due mainly 
to motion of 
the equator 
as liroved 
by the con- 
stancy of the 
of stars. 

Fig. 55. — Conical Motion of Earth's Axis 

of the pole 
of the 
around the 
pole of the 



In consequence of this shift of the axis the pole of the 
heavens, i.e.^ that point in the sky to which the line CA happens 
to be directed at any time, describes a circle around the pole of 
the ecliptic in a period of about twenty-five thousand eight 
hundred years (360° -^ 50'^2). While the pole of the ecliptic 
has remained almost fixed among the stars, the pole of the 

Ir^JPole of Rotation 

. a Cygni 

\ Former 
Pole Star 

a Cepliei 
84100 A. D. 

4600 B.C. 

5 Cygni 

77n o LyrcB 
Fig. 56. — Precessional Path of the Celestial Pole 

equator has traveled many degrees since the earliest observa- 
tions. Fig. ^Q shows approximately its path among the northern 
Path of the constcUations ; not exactly^ of course, on account of the continual 
pole among g;[igrht shifting^ of the plane of the earth's orbit, which makes the 

the stars. ^ . 

pole of the ecliptic move about a little, so that the center of the 
" precessional circle " is not an absolutely fixed point. 



Reckoning back about four thousand six hundred years, we see from the 
figure that a Draconis was then the pole-star, and about five thousand six Former and 
hundred years hence a Cephei will take the office. The circle passes not future pole- 
very far from Yega on the opposite side from the present pole-star, so that ^ ^^^* 
about twelve thousand years from now Vega (a Lyrae) will be the pole- 
star, — a splendid one but rather inconveniently far from the pole. 

N.B. — This precessional motion of the celestial pole must not be confounded 
with the motions of the terrestrial pole which cause the variations of latitude. 

Fig. 57. 

167. Displacement of the Signs of the Zodiac. — Another Effect of 
effect of precession is that the signs of the zodiac (Sec. 157) Precession 

•^ _ "^ ^ ' on the signs 

do not now agree with of the 

the constellations of ^hich /T^m — t^^ zodiac, 

they bear the name. The 
sign of Aries is now in 
the constellation of Pisces, 
and so on. In the last 
two thousand years each 
sign has backed bodily, 
so to speak, into the con- 
stellation west of it. 

Great changes have 
taken place also in the 
apparent position of other constellations in the sky. Six thou- 
sand years ago the Southern Cross was visible in England and 
Germany, and Cetus never rose above the horizon. 

168. Physical Cause of Precession. — The physical cause of Mechanical 
this slow conical motion of the earth's axis was first explained explanation 

^ ^ of preces- 

by Newton, and lies in the two facts that the earth is not sion. 
exactly spherical, but has, so to speak, a protuberant ring of 
matter around its equator, — the equatorial hulge^ — and that 
the sun and moon act upon this ring, tending (but not able) to 
draw the plane of the equator into coincidence with the plane 
of the ecliptic, as a magnet tends to draw the plane of an iron 
ring into line with its pole. 

Precession illustrated by the 



tion of 
by the 

Why preces- 
sion is so 

Equation of 
the equinox 
due to varia- 
tions in the 
force which 

If it were not for the earth's rotation, this action of the sun 
and moon would actually bring the two planes of the equator 
and ecliptic into coincidence ; but since the earth is spinning 
on its axis, we get the same result that we do with the whirling 
wheel of a gyroscope by hanging a weight at one end of its 
axis (Fig. 57). We then have a combination of two rotations at 
right angles to each other, — one the whirl of the wheel, the 
other the "tip" which the weight tends to give the axis. The 
resultant effect — very surprising when the experiment is seen 
for the first time — is that the axis of the wheel, instead of 
tipping, maintains its inclination unchanged, but moves around 
eonieally like the axis of the earth, as shown in Fig. b^. Any 
force tending to change the direction of the axis of a whirling 
body produces a motion exactly at right angles to its own 

Compared with the mass of the earth and its energy of rota- 
tion, this disturbing force is very slight, and consequently the 
rate of precession is extremely slow. If the earth were spher- 
ical, there would be no precession. If it revolved on its axis 
more slowly, precession would be more rapid, as it would be 
also if the sun and moon were larger or nearer, or if the 
obliquity of the ecliptic were greater, not exceeding 45°. 

The moon, being nearer than the sun, is much the more 
effective of the two in producing the precession. 

169. Equation of the Equinox. — The force which tends to 
pull the equator towards the ecliptic continually varies. When 
the sun and moon are crossing the celestial equator the action 
becomes zero — twice a j^ear for the sun, twice a month for the 
moon. Moreover, as we shall see (Sec. 192), the maximum 
declination attained by the moon during the month changes all 
the way from 18° 07' to 28° 47', and its effect in producing 
precession varies correspondingly. As a consequence there is, 
superposed upon the mean precession of the equinoxes, a small 
periodic variation in its rate, producing the equation of the 


equinox^ a slight advance or recession of the equinox from 
its mean place never amounting to more than a few seconds 
of arc. 

170. Nutation. — This is a slight motion of the pole of the 
equator alternately towards and from the pole of the ecliptic, — Nutation a 
a "nodding," so to speak, of the pole. In most positions of slight peri- 
the sun or moon with respect to the equator, there is, in addi- motion of 
tion to the "tipping" force, a slight thwartwise action, tending tiiepoie 

to accelerate or retard the precession, just as if one should gently ^^^^ ^-^^ 
draw horizontally the weight W ?it the end of the axis (Fig. 57). pole of the 
The actual effect in this case is not to change the rate of preces- ^^ ^^^ ^^' 
sion in the least, but to alter the inclination of the axis. This 
causes a nutation amounting to about 9".2 as a maximum and 
running through its principal changes in nineteen years, — the 
period in which the nodes of the moon's orbit complete their 
circuit (Sec. 192). 

171. Aberration of Light. — Aberration^ is the apparent dis- Aberration 
placement of a heavenly body., due to the combination of the orbital ^ ^^ ^^ 
velocity of the earth with the velocity of light. 

The fact that light is not transmitted instantaneously, but 
with a finite velocity, causes the displacement of an object 
viewed from any moving station, unless the motion is directly 
towards or from that object. If the motion of the observer 
is slow as compared with the speed of light, this displace- 
ment is insensible; but the earth moves swiftly enough (about 
18i miles per second) to make it easily observable with modern 
instruments. The direction in which we point our telescope to 
observe a star is usually not the same as if we were at rest, and 
the angle between the two directions is the star's aberration 
at the moment. 

We may illustrate this by considering what would happen in illustration 
the case of falling^ raindrops observed by a person in motion, f^"^"^ ^]^^ 

^ ^ t/ 1. behavior 

1 It was first discovered and explained in 1726 by Bradley, then the English ^ . ^ "^^ 

^ '' . raindrops. 

Astronomer Royal. 



giving tlie 
in terms 
of the 

Suppose the observer standing with a tube in his hand while 
the drops are falling vertically. If he wishes to have the drops 
descend through the tube without touching the side, he must 
obviously keep it vertical so long as he stands still ; but if he 
advances in any direction, the drops will strike his face and 
he will have to draw back the bottom of the tube (Fig. 58) by 
an amount which equals the advance he makes during the time 
Avhile a drop is falling through it ; z.e., he must incline the tube 
forward at an angle, a, which depends both upon the velocity 
of the raindrop and the velocity of his own motion, so that when 
the drop which entered the tube at B reaches A' the bottom of 

the tube will be there also. 

This angle is given by the 

{^ /.. equation 

tan a = — , 

Fig. 58. — Aberration of a Eaindrop 

in which V is the velocity of the 
drop, and u the velocity of the 
observer at right angles to V. 

It is true that this illustration is 
not a demonstration, because light 
does not consist of particles coming 
towards us, but of ivaves transmitted 
through the ether of space. But it has been shown (though the proof is 
by no means elementary) that, within very narrow limits, if not exactly, 
the apparent direction of tha motion of a wave is affected in precisely the 
same way as that of a moving projectile. 

Tiie constant 172. The Constant of Aberration. — By the discussion of thou- 

of aberra- sands of observations upon stars during the past fifty years, 

it is found that the maximum aberration of a star — the same 

for all stars — is about 20''. 5,^ which is called the Constant of 

Aberration. This maximum displacement occurs, of course. 

1 This value is uncertain by at least 0.02 or 0.03 of a second, 
nomical Congress at Paris in 1896 adopted the value 20'\47. 

The Astro- 


whenever the sun's motion is at right angles to the line drawn 
from the earth to the star, always twice a year. 

A star at the pole of the ecliptic is, however, permanently Aberra- 
in a direction perpendicular to the earth's motion, and will ^i^nai orbit 


therefore always be displaced by the same amount of 20 '.5, but annually by 
in a direction continually changing. It therefore appears to ^^^h star. 
describe during the year as its " aberrational orbit " a little circle 
4-1" in diameter. 

A star on the ecliptic (latitude 0°) appears simply to oscillate 
back and forth in a straight line 41" long. 

Between the ecliptic and its pole the aberrational orbit is an 
ellipse having its major axis parallel to the ecliptic and always 
41" long., while its minor axis depends upon the star's latitude, 
yS, and always equals I^V^ sin 13. 

There is also a very slight diurnal aberration due to the rotation of the 
earth, its amount dej^ending on the observer's latitude and ranging from 
0''.31 at the equator to zero at the pole. 

173. Determination of the Earth* s Orbital Velocity and the Distance of 
Mean Distance of the Sun by Means of Aberration. —From ^^^e sun 


Sec. 171, tan a = — , which gives u = V tan a, u in this case i^y means of 

V aberration. 

being the velocity of the earth in its orbit and V the velocity 
of light, while a is the constant of aberration. The experi- 
ments of Michelson and Newcomb (Physics, p. 604) (con- 
firmed in 1900 by Perrotin's experiments at Paris by a different The meas- 
method) make F equal 186330 miles a second, with a probable ^^^^edveioc- 

/ 1 or ity of light. 

error of about 25 miles. We have, therefore, u, the velocity 
of the earth in its orbit, equals 186330 x tan 20''.47 = 18.5 

The circumference of the orbit, regarded as circular (which Resulting 
in the case of the earth involves no sensible error), is found ^^^^^® ^^^" 

the sun's 

by multiplying this velocity, 18.5, by the number of mean distance, 
solar seconds in the sidereal year (Sec. 182). Dividing this 



Amount of 

Solar day 
about four 
longer than 
the sidereal. 

solar days 
are of 

The fictitious 

of the equa- 
tion of time. 

circumference by 2 tt, we find the radius of the orbit, or the 
mean distance of the sun, to be very nearly 92 900000 miles. 

The uncertainty of the constant of aberration affects the 
distance proportionally, by perhaps 100000 miles. Still the 
method is one of the very best of all that we possess for deter- 
mining the value of " the astronomical unit." 

174. Solar Time and the Equation of Time. — Since the sun 
makes the circuit of the heavens in a 3^ear, moving always 
towards the east, the solar day, as has been already explained 
in a preceding article, is about four minutes longer than the 
sidereal day, the difference amounting to exactly one day in a 
year; ^.e., while in a sidereal year there are 366i (nearly) sidereal 
days, there are only 365? solar days. Moreover, the sun's 
advance in right ascension between two successive noons varies 
materially, so that the apparent solar days are not all of the 
same length. December 23 is fifty-one seconds longer than 
September 16. 

Accordingly, as already explained (Sec. 98), mean time has 
been adopted, which is kept by a '^ fictitious," or " mean," sun 
moving uniformly in the equator at the same average rate as 
that of the real sun in the ecliptic. The hour angle of this 
mean sun is the local mean time, or clock time, while the hour 
angle of the real sun is the apparent, or sun-dial, time. 

The Equation of time is the difference between these two 
times reckoned as plus when the sun-dial is slower than the 
clock and minus when it is faster; i.e., it is the correction which 
must be added (algebraically) to apparent time in order to get 
mean time, and this is simply equal to the difference between 
the right ascensions of the fictitious sun and the true sun; so 
that, calling the equation of time E, we may write ^ = a^ — a^^, 
in which a^ is the right ascension of the true sun and a^ the 
right ascension of the mean sun. When a^ is greater than a^ 
the real sun comes to the meridian later than the true sun, and 
the sun-dial is slow of mean time. 


The principal causes of this difference are two : 

(1) The variable motion of the sun in the ecliptic, due to the Causes of 
eccentricity of the eartNs orhit. ^^^® equation 

, , . . 7 . . of time. 

(2) The obliquity of the ecliptic. 

175. Effect of the Eccentricity of the Earth's Orbit. — Near 
perihelion, which occurs about December 31, the sun's eastward 
motion on the ecliptic is most rapid. At this time, accordingly, 
the apparent solar days, for this reason, exceed the sidereal by 

more than the average amount, making the sun-dial days longer Effect of the 
than the mean. The sun-dial will therefore lose time at this ^"^^^i^^^^^ 

velocity of 

season and will, so far as this cause is concerned, continue to the earth in 
do so until the motion of the sun falls to its average value, as ^^i^e^'ent 

•n 1 1 P 1 ^ • • ^ nc partS Of itS 

it will at the end oi three months ; at this time the difference orbit, pro- 
will have amounted to about 7| minutes. Then the sun-dial f^ucingan 
will gain until aphelion, and at that time the clock and sun- of ±7^^ ^ 
dial will once more agree. During the remaining half of the minutes. 
year the action will be reversed. The equation of time, there- 
fore, so far as due to this cause only, is about + 7f minutes in 
the spring, and — 7| in the autumn. 

176. Effect of the Obliquity of the Ecliptic. — Even if the Equation 
sun's motion in longitude [i.e., along the ecliptic) were uniform, ^^|^etothe 
its motion in right ascensio7i would be variable. If the true the ecliptic. 
and fictitious suns were together at the vernal equinox, one 
moving uniformly in the ecliptic and the other in the equator, 

they would indeed be together (i.e., have the same right ascen- Uniform 
sions) at the two solstices and at the other equinox, because it "^°^^^^^^ 

' -'- sun on the 

is just 180° from equinox to equinox and the solstices are ecliptic does 
exactly half-way between them ; but at any point between the ^^^* ^^^® ^ 

. . . . . .pp uniform 

solstices and equinoxes their right ascensions would differ. motion in 

This is easily seen by taking a celestial globe and marking ^'[s^^^ ascen- 
on the ecliptic the point m (Fig. 59), half-way between the 
vernal equinox J^ and the summer solstice C, and also marking 
a point n on the equator 45° from the equinox. It will be seen 
at once that the former point is west of n, the difference of 




Equation of 
time due to 
this cause 
about + 10" 
six weeks 
before each 
and -lO-" 
six weeks 
after it. 

of tlie two 
of the equa- 
tion of time, 
sliowing tlie 
total result 
and effect. 

right ascension being m'n^ so that m in the apparent diurnal 
revolution of the sky will come first to the meridian.^ In other 
words, about six weeks after each equinox, when the sun is half- 
way between the equinox and the solstice, the sun-dial, so far 
as the obliquity of the ecliptic is concerned^ is faster than the clock, 
and this component of the equation of time is minus, amounting 
to nearly ten minutes. Of course, the same thing holds, with 
the necessary changes, for the other quadrants. 

If the ecliptic be divided into equal portions from JE to C 
and hour-circles be drawn from P through the points of division, 
it is clear that near U tlie portions of the ecliptic are longer 

than the corresponding portions 
of the equator. On the other 
hand, near the solstice C the 
arc of the ecliptic is shorter 
than the corresponding arc of 
the equator, on account of the 
divergence of the hour-circles as 
they recede from the pole. 

177. Combination of the 
Effects of the Two Causes. — 
We can represent the two com- 
ponents of the equation of time and the result of their combi- 
nation by a graphical construction (Fig. 60). 

The central horizontal line is a scale of dates one year long, 
the months being indicated at the top. The dotted curve shows 
that component of the equation of time which is due to the 
eccentricity of the earth's orbit. In the same way the hrokeyi- 
line curve denotes the effect of the obliquity of the ecliptic. 
The heavy-line curve represents the combined effect of the two 

Fig. 59 

1 In the figure the observer is supposed to be looking at the globe /ro??i the 
west, E, the vernal equinox, being at the west point of the horizon, ECA is 
the ecliptic, its pole being JT; while EQAT is the celestial equator, its pole 
(of diurnal rotation) being P. 








1 - 1 1 

Ss oS c^S 

+ + 

CnS 1 


































• / 


















^ %^. 

>* ^W 









1 N 




















) ^ 



/ ; 



























/ • 











\ • 






^ • 





, + 


+ 4 






o, q 

? - ^ 





causes, its ordinate at each point being made equal to the 
algebraic sum of the ordinates of the other two curves. The 
heavy-line curve is carefully laid out from the Nautical Almanac 
for 1902 (a mean year in the "leap-year cycle") and will give 
the equation of time for any date during the next fifty years 
within less than half a minute ; not exactly, because from year 
to year the equation of time for any day of the month varies 

Autumnal Equinox 




Vernal Equinox 

Fig. 61. — The Seasons 

Other causes 
slightly to 
the equation 
of time. 

Dates when 
of time 

a few seconds. The small rectangles reckoned horizontally 
represent fiftee7i-day intervals ; vertically, intervals of five 

The two causes discussed above are only the principal ones. 
Every perturbation suffered by the earth slightly modifies the 
result, but all other causes combined never affect the equation 
of time by as much as ten seconds. 

The equation of time becomes zero four times yearly, as will 
be seen from the figure, — about April 15, June 14, Septem- 
ber 1, and December 24; but the dates vary a little from year 
to year. 



178. The Seasons. — The earth in its orbital motion keeps its 
axis nearly parallel to itself for the same mechanical reason 
that a spinning globe maintains the direction of its axis unless 
disturbed by some outside force, — very prettily illustrated by 
the gyroscope. Since this axis is not ^perpendicular to the 
plane of its orbit, the poles of the earth vary in their presenta- 
tion to the sun, as shown in Fig. 61. At the two equinoxes, 
March 21 and September 22, the plane of the earth's equator 
passes through the sun, so that the circle which divides day 
from night upon the earth passes through the pole, as shown in 
Fig. 62, B^ and day and night are then everywhere equal. On 
June 21 the earth is so situated that its north pole is inclined 
towards the sun by about 23^°, as shown in Fig. 62, A, The 
south pole is then in the unillumi- 
nated half of the globe, while the 
north pole receives sunlight all day 
long ; and in all portions of the north- 
ern hemisphere the day is longer 
than the night, and vice versa in the 
southern hemisphere. At the time 
of the winter solstice these condi- 
tions are reversed and the south pole has perpetual sun- 
shine. On the equator day and night are equal at all times 
of the year, and there are no seasons in the proper sense of 
the word. 

The midnight sun and other phenomena in the neighborhood 
of the pole have already been discussed (Sec. 36). 

179. Effects on Temperature. — The changes in the duration 
of insolation (exposure to sunshine) at any place involve changes 
of temperature and of other climatic conditions which produce 
the seasons. Taking as a standard the average amount of heat 
received from the sun in twenty-four hours on the day of the 
equinox, it is clear that the surface of the soil at any place in 
the northern hemisphere will receive each twenty-four hours 

of north and 
south poles 
of the earth 
to the sun. 

Fig. 62. — Position of Pole at 
Solstice and Equinox 

Station in 
more than 
the average 
amount of 
heat in a 
day when 
sun is north 
of equator. 



Two reasons ; 
day is 
then longer 
than the 
night, and 
during the 
day the 
sun's mean 
altitude is 

effect of 
sun's rays 
varies with 
the sine of 
sun's alti- 
tude, modi- 
fied hy 

amount of 
heat lost in 
a day equals 
that re- 

more than the average of heat whenever the sun is north of the 
celestial equator, and for two reasons : 

(1) Sunshme lasts more than half the day. 

(2) The mean altitude of the sun while above the horizon is 
greater than at the time of the equinox. 

Now the more obliquely the rays strike the less heat they 
bring to each square inch of the surface, as is obvious from 
Fig. 63. A beam of sunshine having a cross-section, ABCJD, 
is spread over a larger area when it strikes obliquely than when 
vertically, its heating efficiency being in inverse ratio to the 
surface over which the heat is distributed. If Q is the amount 
of heat per square meter of area brought by the rays when fall- 
ing perpendicularly, as on the sur- 
face AC, then on Ac, on which it 
Cy' j^ X \^^1 strikes at the angle li (equal to the 

sun's altitude), the amount per 
square meter will be only Q X sin h. 
Moreover, this difference in favor 
"^A of the more nearly vertical rays is 

Fig. (53. -Effect of Sun's Elevation exaggerated by the absorption of 
on Amount of Heat imparted to heat in the atmosphere, since rays 

that are nearly horizontal have to 
traverse a much greater thickness of air before reaching the 

For these two reasons, therefore, at a place in the northern 
hemisphere the mean temperature of the day rises rapidly as 
the sun comes north of the equator, thus causing summer. 

180. Time of Highest Temperature. — We receive the most 
heat in twenty-four hours at the time of the summer solstice ; 
but this is not the hottest time of the season for the obvious 
reason that the weather is still getting hotter, and the maxi- 
mum will not be reached until the increase ceases; i.e., not 
until the amount of heat lost in twenty-four hours equals that 
received, which occurs in our latitude about August 1. For 


similar reasons the minimum temperature of winter occurs 
about February 1. 

Since the weather is not entirely " made on the spot where 
it is used," but is much influenced by winds and currents that 
come from great distances, the actual date of the maximum tem- 
perature at any particular place cannot be determined beforehand 
by mere astronomical considerations, but varies considerably 
from year to year. The great differences between the seasons 
of different years are as yet mostly without explanation. 

181. Difference between Seasons in Northern and Southern Effect of the 
Hemispheres. — Since in December the distance of the earth eccentricity 

of the 

from the sun is about three per cent less than it is in June, the earth's orbit 
earth as a whole receives hourly about six per cent more heat in producing 
in December than in June, the heat received varying inversely between the 
as the square of the distance. For this reason the southern seasons in 
summer, which occurs in December and January, is hotter than ^^^ south- 
the northern. It is, however, seven days shorter, because the ernhemi- 
earth moves more rapidly in that part of its orbit. The total ^^ ^^®^* 
amount of heat per acre received during the whole summer 
is therefore sensibly the same in each hemisphere, the short- 
ness of the southern summer making up for its increased 

The southern winter^ however, is both longer and colder than Question 
the northern, and it has been vigorously maintained by certain wi^e|^i^er the 
geologists that, on the whole, the mean annual temperature of period can 
the hemisphere which has its winter at the time when the earth ^® explained 
is in aphelion is lower than the opposite one. It has been effect. 
attempted to account for the glacial epochs in this way, but the 
explanation is very doubtful. 

On account of the motion of the apsides of the earth's orbit 
(Sec. 164) the present state of things will be reversed in about 
ten thousand years; the perihelion will then be reached in 
June^ and the northern summer will then be the shorter and 
the hotter one. 


The three 182. The Three Kinds of Year. — Three different kinds of 

kinds of "year" are now recognized, — the sidereal^ the tropical or equi- 
noctial^ and the anomalistic. 
The sidereal The Sidereal Year, as its name implies, is the time occu- 
year,— ^^^^^ j^^ ^1^^ g^j-^ ^^ apparently completing the circuit of the 

days. heavens from a given star to the same star again. Its length is 

365^6^9™9« of mean solar time (365^25636). 

From the mechanical point of view this is the true year ; i.e., 

it is the time occupied by the earth in making one complete 

revolution around the sun from a given direction in space to 

the same direction again. 

The tropical The Tropical Year is the time included between two suc- 

?fi? 94910 cessive passages of the vernal equinox by the sun. On account 

days. of preccssion (Sec. 165) the equinox moves yearly 50''. 2 towards 

the west, so that the tropical year is shorter than the sidereal, 

its length being 365^^5M8™45^5 (365^^24219). Its length was 

determined by the ancients with considerable accuracy, as 365i 

days, by means of the gnomon ; they noted the dates at which 

the noonday shadow was longest (or shortest), i.e., the date of 

the solstice. 

Since the seasons depend on the sun's place with respect to 
The tropical the equiuox, the tropical year is the year of chronology and 
year the civil reckoning. Whenever a period of so many years is spoken 
chronology, of wc always Understand tropical years, unless otherwise dis- 
tinctly indicated. 
The anoma- The third kind of year is the Anomalistic Year, — the time 
listic year,— between two successive passages of the perihelion. Since the 
(Jays. liiie of apsides of the earth's orbit moves eastward about 11' a 

year (Sec. 164), this kind of year is nearly five minutes longer 
than the sidereal, its length being 365^6^13™48^ (365*^.25958). 
It is little used, except in calculations relating to perturba- 
Natural 183. The Calendar. — The natural units of time are the day, 

time units, nionth, and year. The day is too short for convenience in 


dealing with considerable periods, — such as the life of a man, 
for instance ; and the same is true even of the month, so that 
for all chronological purposes the tropical year — the year of 
the seasons — has always been employed. At the same time, so 
many religious ideas and observations have been connected with 
the changes of the moon that there was long a constant struggle to 
reconcile the month with the year. Since the two are incommen- 
surable, no really satisfactory solution is possible, and the modern 
calendar of civilized nations entirely disregards the moon. 

In the ancient times the calendar was in the hands of the priesthood 
and was predominantly lunar, the seasons either being disregarded or kept Lunar 
roughly in place by the occasional intercalation or dropping of a month, calendars. 
The principal Mohammedan nations still use a purely lunar calendar having 
a year of twelve lunar months containing alternately 354 and 355 days. In 
their reckoning, therefore, the months and their religious festivals fall con- 
tinually in different seasons, and their calendar gains on ours about one 
year in thirty-three. 

184. The Julian Calendar. — When Julius Caesar came into 
power he found the Roman calendar in a state of hopeless con- 
fusion. He therefore sought the advice of the Alexandrian 
astronomer Sosigenes, and in accordance with his suggestions 
established (45 B.C.) what is known as the Julian calendar^ The Julian 
which still, either untouched or with a trifling modification, ^^i^^^ar .- 

. every fourth 

continues in use among all civilized nations. He discarded all year a leap- 
consideration of the moon, and adopting 8664 days as the true y^^^- 
length of the year, he ordained that every fourth year should 
contain 366 days, the extra day being inserted by repeating the 
sixth day before the calends of March, whence such a year is 
called bissextile. He also transferred to January 1 the begin- why leap- 
ning of the year, which until then had been in March (as is year is called 


indicated by the names of several of the months, as September, ^iie." 
i.e.., the seventh month, etc.). 

Csesar also took possession of the month Quintilis, naming 

it July after himself. His successor, Augustus, in a similar 


ma:n^ual of astronomy 

ness of the 

The Grego- 
rian calen- 
dar. Correc- 
tion made, 
and error 
from accu- 
mulating by 
new rule 

Adoption of 
the Grego- 
rian calen- 
dar in Eng- 
land in 1752. 

manner appropriated the next month, Sextilis, calling it August^ 
and to vindicate his dignity and make his month as long as his 
predecessor's he added to it a day stolen from February. 

The Julian calendar is still used unmodified in Russia and 
by the Greek Church generally. 

185. The Gregorian Calendar. — The true length of the tropi- 
cal year is not 365i days, but 365'^5^'48M5^5, leaving a differ- 
ence of ll'^ltt^S by which the Julian year is too long. This 
amounts to a little more than three days in four hundred years. 
As a consequence, in the Julian calendar the date of the vernal 
equinox comes earlier and earlier as time goes on, and in 1582 
it had fallen back to the 11th of March instead of occurring 
on the 21st, as it did at the time of the Council of Nice, A.D. 325. 
Pope Gregory, therefore, under the advice of the distinguished 
astronomer Clavius, ordered that the calendar should be cor- 
rected by dropping ten days, so that the day following Oct. 4, 
1582, should be called the 15th instead of the 5th ; and further, 
to prevent any future displacement of the equinox, he decreed 
that thereafter only such century years should be leap-years as 
are divisible by 400, (Thus, 1700, 1800, 1900, 2100, and so on, 
are not leap-years, while 1600 and 2000 are.) 

186. The change was immediately adopted by all Catholic 
countries, but the Greek Church and most Protestant nations 
refused to recognize the Pope's authority. It was, however, 
finally adopted in England by an act of Parliament, passed in 
1751, providing that the year 1752 should begin on January 1 
(instead of March 25, as had long been the rule in England) 
and that the day following Sept. 2, 1752, should be reckoned 
as the 14th instead of the 3d, thus dropping eleven days. 

The change was bitterly opposed by many, and there were riots in conse- 
quence in various parts of the country, especially at Bristol, where several 
persons were killed. The cry of the people was, " Give us back our fort- 
night," for they supposed they had been robbed of eleven days, although 
the act of Parliament was carefully framed to prevent any injustice in the 
collection of interest, payment of rents, etc. 


At present, since the years 1800 and 1900 were leap-years in 
the Julian calendar and not in the Gregorian, the difference 
between the two calendars is thirteen days ; thus, in Russia the Present 
22d of June is reckoned the 9th, but in that country both dates d^ff^^'e^^'e of 

'^ the two 

are ordinarily used for scientific purposes, so that the date would calendars is 
be written June -f^. thirteen 

When Alaska was annexed to the United States the official ^iii remain 
date had to be changed by only eleven days, one day being ^o until the 
provided for in the alteration from the Asiatic reckoning to the 
American (Sec. 111). 

187. The Metonic Cycle and Golden Number. — In establishing 
a relation between the solar and lunar years, the discovery of 

the so-called lunar (or Metonic) cycle by Meton, about 433 B.C., The Metonic 
considerably simplified matters. This cycle consists of 235 ^X^^~ 

^ ^ ^ nineteen 

synodic months (from new moon to new again), which is very years— very 
approximately equal to nineteen common years of 365i days, ^^^ariy equal 
The calendar for the phases of the moon is, therefore (with very months. 
rare exceptions), the same for any two years nineteen years apart; 
i.e., the calendar of the phases of the moon, and of all ecclesias- 
tical holidays which depend upon them (Easter, etc.), is the 
same for 1901 as for 1872 and 1920. But the dates are liable 
to a shift of a single day, according to the number of leap-years 
which intervene. This cycle is still employed in the ecclesias- 
tical calendar in determining the time of Easter. 

The golden number of a year is its number in this Metonic The "golden 
cycle and is found by adding 1 to the date number of the year ^^^"^^^i"' 
and dividing by 19. The remainder, unless zero, is the golden 
number. If it comes out zero, 19 is taken. Thus, the golden 
number for the year 1902 is 3. 

188. The Julian Period and Julian Epoch. — The Julian Period The Julian 
consists of 7980 Julian years (28 x 19 x 15), each contain- P^"^^^^ .^"'^ 

, . epoch mtro- 

ing exactly 365? days, and its starting-point, or Epoch, is ducedhyj. 
Jan. 1, 4713 B.C., — the Julian date of Jan. 1, a.d. 1, being Scaiiger. 
J.E. 4714. 


The system was proposed by J. Scaliger in 1582 as a uni- 
versal harmonizer of the different systems of chronological 
reckoning then in use, and its adoption has brought order out 
of confusion. It is extensively employed in astronomical calcu- 
lations, the date of any phenomenon being expressed beyond all 
Julian date ambiguity either by the (Julian) year and day, or still more 
and day simply by " day number " so and so of the Julian era. Thus, the 

numbers. l j j j 

date of the solar eclipse of Aug. 9, 1896, is J.E. 6609, 222d day, 
or simply Julian day 2 413781 ; and this is perfectly definite to 
every astronomer of whatever nation, — American, Russian, 
Arabian, or Chinese. 

The number of days between any two events, even centuries 
apart, is at once found by merely taking the difference between 
their Julian day numbers. 

The Almanac gives for each year its Julian number, and also the Julian 
day number for January 1 of that year. 

1900 is Julian year 6613. Jan. 1, 1900, is Julian day 2 415021. 
1902" " " 6615. " 1,1902," " " 2 415751. 

March 10, 1902, " " " 2 415820, etc. 

For a fuller explanation of the considerations on which this system 
of reckoning is founded, the reader is referred to Herschel's Outlines of 
Astronomy, Art, 924. 


1. What is the meridian altitude of the sun at Princeton (Lat. 40° 21') 
on the day of the summer solstice ? 

2. What is the sun's approximate right ascension at that time? 

3. On what days during the year will the sun's right ascension be 
approximately an even hour {i.e., hours, 2 hours, 4 hours, etc.)? 

4. On what days will it be an odd hour? 

5. What is the (approximate) sidereal time at 10 p.m. on May 12 ? 

Ans. 13^26™. 

6. At what time will Arcturus (R.A. = 14^il0ii^) come to the meridian 
on August 1? Ans. About 5^2611^ p.m. 

7. About what time of night is Mizar (R.A. = 131^20™) vertically under 
the pole on October 10? Ans. Midnight. 


8. In what latitude has the sun a meridian altitude of 80° on June 21 V 

A71S. + 83° 27'. 

9. What are the longitude and latitude (qelestial) of the north celestial 
pole? Ans. Long. 90°, Lat. 66° 33'. 

10. What are the right ascension and declination of the north pole of 
the ecliptic? Ans. R.A. 18^ Dec. 66° 33'. 

11. What are the greatest and least angles made by the ecliptic with 

the horizon at New York (Lat. 40° 43') ? 

< Max 72° 44' 

A?is. (90° - 40° 43') ± 23° 27' = ] '^ ^^ • 

^ ^ i Min. 25° 50'. 

12. Does the vernal equinox always occur on the same day of the 
month ? If not, why not ? How much can the date vary ? 

13. Will the ephemeris oi the sun for one year be correct for every 
other year, and, if not, how much can it be in error ? 

A71S. A difference of 1|- days' motion of the sun is possible ; as, for 
instance, between 1897 and 1903, the leap-year being omitted in 1900. 

14. When the sun is in the sign of Cancer in what constellation is he? 

15. What obliquity of the ecliptic would reduce the width of the tem- 
perate zone to zero ? 

16. At a place west of Philadelphia an observer finds that his local 
apparent time on October 1, as determined from the sun by sextant, was 
8^30^ slow of eastern standard time. The equation of time on that date 
is - 10"i8s. What was his longitude from Greenwich ? Ans. 5iil8m38s. 

17. At what standard time will the sun come to the meridian on 
March 21 at Boston (Long. 4M4™ west of Greenwich), the equation of 
time being + 7^288 ? Ans. 11^51^288. 

18. When the equation of time is 1(5 minutes, as it is on November 1, 
how does the forenoon from sunrise till 12 o'clock compare in length with 
the afternoon from 12 o'clock till sunset? 

19. Why do the afternoons begin to lengthen about December 8, a fort- 
night before the winter solstice ? 

20. There were five Sundays in February, 1880. The same thing has 
not occurred since, and will not until when ? Ans. 1920. 

21. What was the Russian date corresponding to Feb. 28, 1900, in our 
calendar ? What corresponding to May 1 of the same year ? 

Ans. February 16; April 18. 



of the 
moon in 

motion of 
tlie moon 
among the 

The Moon's Orhital Motion and the Month — Distance, Dimensions, Mass, Density, 
and Force of Gravity — Rotation and Librations — Pliases — Liglit and Heat — 
Pliysical Condition — Telescopic Aspect and Peculiarities of the Lunar Surface 

189. Next to the sun, the moon is the most conspicuous and 
to us the most important of the heavenly bodies, — in fact, the 
only one except the sun which exerts the slightest influence 
upon human life. If the stars and planets were all extin- 
guished, our eyes would miss them, and that is all ; but if the 
moon were annihilated, the interests of commerce would be 
seriously affected by the practical cessation of the tides. She 
owes her conspicuousness and importance, however, solely to her 
nearness, for she is really a very insignificant body as compared 
with the stars and the planets. 

And yet, astronomically, she perhaps ranks highest among 
the heavenly bodies. The very beginnings of the science seem 
to have originated in the study of her motions and of the differ- 
ent phenomena which she causes, such as the eclipses and the 
tides; and in the development of modern theoretical astron- 
omy the "lunar theory," with the problems it raises, has been 
perhaps the most fertile field of discovery and invention. 

190. The Moon's Apparent Motion; Definition of Terms, etc. 
— One of the earliest observed of astronomical phenomena 
must have been the eastward motion of the moon with reference 
to the sun and stars and the accompanying changes of phase. 
If we note the moon to-night as very near some conspicuous 
star, we shall find her to-morrow night at a point about 13° 
farther east, and the next night as much farther still; she 



makes a complete circuit of tlie heavens, from star to star 
again, in about 27-g- days. In other words, she " revolves 
around the earth" in that time, while she accompanies us in 
our annual journey around the sun. 

Since the moon moves eastward among the stars so much The moon's 
faster than the sun, she overtakes and passes him at ree^ular ^pp^^"®"*^ 

^ _ _ motion with 

intervals ; and as her phases depend upon her apparent position respect to 

with respect to the sun, this interval from new moon to new ^^^ ^^^"• 

moon is specially noticeable and is what we ordinarily under- month, 
stand as the months — technically, the synodic month. 

The Elongation of the moon is her angular distance east or Definitions 

west of the sun at any time. At new moon it is zero, and the ^.^ eionga- 

. . . tion, con- 

moon is said to be in Conjunction. At full moon the elon- junction, 

gation is 180°, and she is said to be in Opposition. In both ®^^- 

cases the moon is in Syzygy^ i.e.^ the sun, moon, and earth are 

ranged nearly along a straight line. When the elongation is 

90° she is said to be in Quadrature. 

191. Sidereal and Synodic Months. — The Sidereal Month The sidereal 

is the time it takes the moon to make her revolution from a ^" y^^° ^^ 

^ months. 

given star to the same star again as seen from the center of 
the earth. It averages 27^^7M3m^55 (27^^32166), but it varies 
some three hours on account of " perturbations." The mean 
daily motion is 360° ^ 27.32166, or 13° 11'. Mechanically 
considered, the sidereal month is the true month. 

The synodic month is the time between two successive con- 
junctions or oppositions, i.e., between successive new or full 
moons. Its average value is 29'U2M1:""2^86, but it varies 
nearly thirteen hours, mainly on account of the eccentricity of 
the lunar orbit. As has been said already, this synodic month 
is what we ordinarily mean when we speak of a " month." 

If M be the length of the moon's sidereal period, E the length Relation 
of the sidereal year, and S that of the synodic month, the three ^®*^®®^ *^^ 

•^ ^ -j sidereal and 

quantities are connected by a very simple relation. — is the sy^^o^ic 

^ J J r jj months. 



fraction of a circumference moved over by the moon in a day. 
Similarly, — is the apparent daily motion of the sun. The 

difference is the amount which the moon gains on the sun 
daily. Now it gains a whole revolution in one synodic month 

of S days, and therefore must daily gain — of the circumference. 


Equation of Hcncc, we have the important equation ~ ~-> "the equa- 

synodic M E S 

tion of svnodic motion," whence S = — 

Number of 
months in a 
year exactly 
one more 
than the 
number of 

The moon's 
path on the 

The nodes. 

t)f the nodes. 

Another way of looking at the matter, leading, of course, to the same 
result, is this : In a sidereal year the number of sidereal months must be 
just one greater than the number of synodic months; the numbers are, 
respectively, 13.369+ and 12.369 + . 

192. The Moon's Path on the Celestial Sphere ; the Nodes and 
their Motion. — By observing the moon's right ascension and 
declination daily with suitable instruments we can map out its 
apparent path, just as in the case of the sun (Sec. 156). It 
turns out to be (very nearly) a great circle inclined to the 
ecliptic at an angle of about 5° 8', but varying 12' each way, 
from 4° m^ to 5° 20'. 

The two points where the path cuts the ecliptic are called 
the nodes^ the ascending node being the one where the moon 
passes from the south side to the north side of the ecliptic. 
The opposite node is called the descending node. (Ancient 
astronomers all lived in the northern hemisphere.) 

The moon at the end of the month never comes back exactly 
to the point of beginning, on account of the so-called " pertur- 
bations," due to the attraction of the sun. 

One of the most important of these perturbations is the 
regression of the nodes. These slide westward on the ecliptic 
in the same manner as the vernal equinox does, but much 
faster, completing their circuit in a little less than nineteen 


years instead of twenty-six thousand. The average time between 

two successive passages of the moon through the same node is 

called the nodical or draconitie month. It is 27.2122 days, — The nodical 

an important period in the theory of eclipses. ordracomtic 

. ^ i . . month. 

When the ascending node of the moon's orbit coincides with 
the vernal equinox the angle between the moon's path and the 
equator has its maximum value of 23° 27' + 5° 8', or 28° 35'; Variation in 
nine and one-half years later, when the descending: node has come *^^® mdma- 

1 -. • 1 ^ , tionofthe 

to the same point, the angle is only 23° 27' — 5° 8', or 18° 19'. moon's path 

In the first case the moon's meridian altitude will range during ^^ *^® ^^^^s- 
the month through about 57°. In the second case the range is 
reduced to 36° 38'. 

193. Interval between the Moon's Successive Transits ; Daily interval 

Retardation of its Rising and Setting. — Owing to the eastward ^^tween 

^ successive 

motion of the moon it comes to the meridian later each day. transits of 
If we call the average interval between its successive transits a *^® moon, — 


" moon day," we see at once that while in the synodic month 

there are 29.5306 mean solar days, there must be just one less 

of these " moon days," since the moon, in the synodic month, 

moves around eastward from the sun to the sun again, thus 

losing one complete relative rotation. 

It follows, therefore, that the length of the " moon day" must 

29 5306 
be 24^ X ' , or 24^50"'.51, the average " daily retarda- How the 

zH.OoOo daily retard- 

tion " being 50^ minutes. It ranges, however, all the way from ation is 
38 minutes to QQ minutes on account of the variations in the its range.* 
rate of the moon's motion in right ascension, — due partly to 
perturbation, but mainly to the oval form of its orbit and its 
inclination to the celestial equator, — variations precisely analo- 
gous to the inequalities of the sun's motion, which produce the 
equation of time (Sec. 174), but many times greater. 

The average retardation of the moon's daily rising and setting 
is also the same 50.51 minutes, but the actual retardation is 
much more variable than that of the transits, depending largely 


ma:n^ual of astro^'omy 

Daily re- 
of moon's 
rising and 
ranges in 
our latitude 
from 23"' to 

One day in 
each month 
when the 
moon does 
not rise. 

on the latitude of the observer. At New York the range is 
from 23 minutes to 77 minutes. In higher latitudes it is still 
greater. Indeed, in latitudes above 61° 20', the moon, when 
it has its greatest possible declination of 28° 47', will become 
circumsolar for a certain time each month and will remain visible 
without setting at all for a whole day or more, according to 
the latitude of the observer. As a consequence of this daily 
retardation it follows that there is always one day in the month 
on which the moon does not rise, and another on which it does 
not set. 

194. Harvest and Hunter's Moon. — The full moon that 
comes nearest the autumnal equinox is known as the harvest- 
moon; the one next following is the hunter^s m,oon. At that 
time of the year the moon while nearly full rises for several 

Fig. 64. — Explanation of the Harvest-Moon 

Harvest and consecutive nights at about the same hour, so that, the moon- 
hunter s lignht evenings last for an unusual length of time. The phe- 

moon. ° . ° ... . 

nomenon is much more striking in Northern Europe than in 
the United States. 

In the autumn the full moon is near the vernal equinox 
(since the sun is at the autunmal) and is in the portion of its 
path which is least inclined to the eastern horizon, where it 
rises. This is obvious from Fig. 64, which represents a celestial 


globe looked at from the east. HN is the horizon, E the east 

point, P the pole, and EQ the equator. If, now, the first of Explanation 

Aries is rising at E^ the line JEJ' will be the ecliptic and will ^^ *^® 

be inclined to the horizon at an angle less than QEH (the ^ ®^°°^®°*'^- 

inclination of the equator) by 231-°. 

If the asce7iding node of the moon's orbit happens to coincide 
with the first of Aries, then, when this node is rising, the moon's 
path will lie still more nearly horizontal than JJ'^ as shown by 
the line MII'^ and the phenomenon of the harvest-moon will 
be specially noticeable. 

195. Form of the Moon^s Orbit. — By observation of the 
moon's apparent diameter, combined with observations of her The moon's 
place in the sky, we can determine the form of her orbit around °f,^^* ^^ . 

^ '^' "^ ellipse with 

the earth in the same way that the form of the earth's orbit a mean 
around the sun was worked out in Sec. 160. The moon's eccentricity 

of about 15. 

apparent diameter ranges from 66' oo", when as near as pos- 
sible, to 29' 24", when most remote. 

The orbit turns out to be an ellipse like that of the earth 
around the sun, but of much greater eccentricity, averaging 
about ^Ig. We say " averaging " because it varies from ylg to 
^Y on account of perturbations. 

The point of the moon's orbit nearest the earth is called the Definition of 
perigee (irepL 7?}), that most remote the apogee (airo 77J), and P^^^see, 

cipO^GGj 3.11 Cl 

the indefinite line passing through these points and continuing apsides. 
to the heavens the line of apsides^ the major axis being that 
portion of this line which lies between perigee and apogee. 
On account of perturbations the line of apsides is in continual Eastward 
motion like the line of nodes, but it moves eastward instead of ^^^^^ "^ 

' ^ the line of 

westward, completing its revolution in about nine years. apsides. 

In her motion around the earth the moon also very nearly 
observes the same " law of areas " that the earth does in her Law of 

orbit around the sun. moon's orbi- 

tal motion. 

196. Method of determining the Size of the Moon^s Orbit, 
i.e., her Distance and Parallax. — In the case of any heavenly 


ma:n^ual of astro^'omy 

tion of the 
moon's dis- 
tance from 
the earth. 

ous merid- 
of the 
tance from 
two stations 
on the same 
hut widely 
separated in 

body one of the first and most fundamental inquiries relates to 
its distance ; until this has been measured we can get no knowl- 
edge of the real dimensions of its orbit, nor of the size, mass, 
etc., of the body itself. The problem is usually solved by 
measuring the ''parallactic displacement" (Sec. 78) due to a 
known change in the position of tlie observer. Many methods 
are applicable in the case of the moon. We limit ourselves to 

a single one, the simplest, 
though perhaps not the most 
accurate, of the different 
methods that are practically 

At each of two observa- 
tories, B and C (Fig. 65), 
on, or very nearly on, the 
same meridian and very far 
apart (Berlin and Cape of 
Good Hope, for instance), 
the moon's zenith-distance, ZB M ?iiid Z'CM^ is observed simul- 
taneously with the meridian-circle. Tliis giA^es in the quadri- 
lateral i?0 CTIi' the two angles Oi?J/ and OCM. The angle i?0(7, 
at the center of the earth, is the difference of the geocentric 
latitudes of the two observatories (numerically their sum). 
Moreover, the sides BO and CO are known, being radii of the 

The quadrilateral can, therefore, be solved by a simple trigo- 
nometrical process. (1) In the triangle BOC we have given BO^ 
OC, and the included angle BOC; hence, we can find the side 
BC and the two angles OBC and OCB. (2) In the triangle 
BCM we now have given BC and the two angles MBC and 
MCB (which are got by simply subtracting OBC from OBM 
and OCB from OCM) ; hence, we can find ^7li"and CM. (3) In 
the triangle OBM or OCM we now know the two sides and 
the included angle at B or C, from which we can find OM^ the 

Fig. 65. — Determination of the Moon's 

about 31400 


moo7is distance from the center of the earth. (4) When OM is 

determined we at once find the horizontal parallax fj'om the 

equation ^ ^ 

Sinn- — -- (Sec. 79). 

197. Parallax, Distance, and Velocity of the Moon. — The 
moon's equatorial horizontal parallax is found to average 3422''. Mean par- 
(57' 2".0), according to Neison, but it varies considerably from ^^^^^ °^ 
day to day on account of the eccentricity of the orbit. Her 57' 02''. 
averap'e distance from the earth is about 60.3 times the earth's Distance 
equatorial radius, or 23884-0 miles^ with an uncertainty of miies. 
10 or 20 miles. 

The maximum and minimum values of the moon's distance Range of 
are given by Nelson as 252972 and 221614 miles. It will be 
noted that the '^ average " distance is not the mean of the two miles 

Knowing the size and form of the moon's orbit, the velocity The moon's 
of her motion is easily computed. It averas^es 2287 miles an ^^^^^^^ 

^ ^ velocity. 

hour, or about 3350 feet per second. Her mean angular velocity 
in the celestial sphere is about 33' an hour, just a little greater 
than the apparent diameter of the moon itself. 

198. Form of the Moon's Orbit with Reference to the Sun. — 
While the moon moves in a small oval orbit around the earth, 
it also moves around the sun in company with the earth. This 
common motion of the moon and earth, of course, does not 
affect their relative motion, but to an observer outside the sys- 
tem looking down upon moon and earth the moon's motion 
around the earth would be a very small component of the moon's 
whole motion as seen by him. 

The distance of the moon from the earth is only about -g-^-Q The moon's 
part of the distance of the sun. The speed of the earth in its P^tiireiative 

*- _ ^ ^ ^ to the sun 

orbit around the sun is also more than thirty times greater than always con- 
that of the moon around the earth ; for the moon, therefore, ^^^'® towards 

. . the sun. 

the resulting path in space is one which deviates very slightly 



from the orbit of the earth and is always co7icave towards the sun, 
as shown in Fig. 66. It is not as shown in Figs. 67 and 68, 
although often so represented. 

If we represent the orbit of the earth by a circle with a radius of 
100 inches (8 feet 4 inches), the moon would deviate from it by only 
one fourth of an inch on each side, crossing it twenty-four or twenty-five 
times in one revolution around the sun, i.e., in a year. 

199. Diameter, Area, and Volume, or Bulk, of the Moon. — 

The mean apparent diameter of the moon is 31' 7''. Knowing 

Fig. 66. — Moon's Path Relative to the Sun 

Size of 
the moon : 
area, and 

Fig. 67 

Erroneous Representation of the Moon's Path 

its mean distance, we easily compute from this (Sec. 12) its real 
diameter, 2 J 63 miles. This is 0.273 of the earth's diameter, — 
somewhat more than one quarter. 

Since the surfaces of globes vary as the squares of their 
diameters, and their volumes as the cubes, this makes the sur- 
face area of the moon equal to 0.0747 (about -^-^) of the earth's, 
and the volume, or bulk, 0.0204 (almost exactly J^) of the 

No other satellite is nearly as large as the moon in compari- 
son with its primary planet. The earth and moon together, as 
seen from a distance, are really in many respects more like a 
double planet than a planet and satellite of ordinary proportions. 

THE MOON^ 175 

When Yenus happens to be nearest us (at a distance of about twenty- Earth and 
five millions of miles) her inhabitants, if she has any, see the earth about moon as 
twice as brilliant as Venus herself at her best appears to us, and the ^^^^ from 
moon, about as bright as Sirius, oscillating backwards and forwards about 
half a degree on each side of the earth. 

200. Mass, Density, and Superficial Gravity of the Moon. — 

The accurate determination of the moon's mass is practically a Determina- 
dif!icult problem. Thoup-h she is the nearest of all the heavenly ^'°^ °^ ^^^® 

■^ ^ ^ ^ ^ ^ ^ mass of the 

bodies, it is far more difficult to lueigJi her than to determine the moon, 
mass of Neptune, the remotest of the planets. There are many 
different methods of dealing with the problem. 

One, perhaps the best, consists in determining the position 
of the eeyiter of gravity, or center of mass, of earth and moon. 
It is this point and not the earth's center which describes 
around the sun what is called the " orbit of the earth." Now 
the earth and moon revolve together around this common center 
of gravity every month in orbits exactly alike in form, but 
differing greatly in size, the earth's orbit being as much smaller 
than the moon's as its mass is greater. 

On account of this monthly motion of the earth's center, there 
results necessarily a "lunar equation," i.e., a slight alternate The "lunar 
eastward and westward displacement in the heavens of every ^^^^^^^^^^ 

^ ^ *^ in the appar- 

object viewed from the earth as compared with the place the cut motion 
object would occupy if the earth had no such motion. In ^^ *^^® ^^^ 

and nearer 

the case oi the stars or the remoter planets the displacement planets, 
is not sensible ; but it can be measured by observing through 
the month the apparent motion of the sun, or better, of one of 
the nearer planets, as Mars or Venus, or the newly discovered 

From such observations it is found that the radius of the 
monthly orbit of the earth's center {i.e., the distance from the 
center of the earth to the common center of gravit}^ of the earth) 

is 2880 miles. This is iust about 7--^ of the distance from the 

*• 82.5 



The moon's earth to the moon, whence we conclude that the mass of the 

mass IS 

8 1.5 

the mass of 
the earth. 

moon IS 

Density of 
the moon 
about three 
fifths the 
density of 
the earth. 


that of the earth. 

Gravity on 
the moon's 
about one 
sixth of 
gravity on 
the earth ; 
important in 
relation to 
the constitu- 
tion of the 

Rotation of 
the moon on 
its axis. The 
period of ro- 
tation equals 
the sidereal 

For other methods of determining the mass of the moon, the reader is 
referred to the General Astronomy, Art. 243. 

201. Since the density of a body is equal to its mass h- volume, 

the density/ of the moon compared with the earth is divided 

by —1 which equals 0.601, or about 3.4 the density of water, the 

earth's density being 5.53. This is a little above the average 
density of the rocks which compose the crust of the earth. This 
low density of the moon is not at all surprising, nor at all incon- 
sistent with the belief that it once formed a part of the earth, 
since, if such were the case, the moon was probably formed by 
the separation of the outer portions of that mass, which would 
be likely to be lighter than the rest. 

The superficial gravity^ or the attraction of the moon for 

bodies at its surface, is mass -^ radius^, i.e.. ——- divided by'0.2732, 

81.5 "^ 

and comes out about one sixth of gravity at the surface of the 
earth. That is, a body weighing six pounds on the earth's sur- 
face would, at the surface of the moon, weigh only one pound 
(by a spring-balance). A man who can leap to a height of 5 feet 
here would reach 30 feet there, and so on.^ 

This is a point that must be borne in mind in connection 
with the enormous scale of the surface structure of the moon. 
Volcanic forces on the moon would throw ejected materials to 
a vastly greater distance than on the earth. 

202. Rotation of the Moon. — The moon rotates on its axis 
once a sidereal month, in exactly the same time as that occupied 
by its revolution around the earth; its day and night, therefore, 
the interval between sunrise and sunset, are each nearly a 

1 But see Sec. 141 for Professor Newcomb's illustration. 


fortnight in length, and m the long run it keeps the same side 
always towards the earth. We see to-day precisely the same 
aspect of the moon as Galileo did in the days when he first 
turned his telescope upon it. 

Many find difficulty in seeing why a motion of this sort should be 
called a " rotation " of the moon, since it is extremely like the motion of a 
ball carried on a revolving crank (Fig. 69). "Such 
a ball," they say, " revolves around the shaft, but p^ 
does not rotate on its own axis." It does rotate, 
however ; for if we mark one side of the ball, we 
shall find the marked side presented successively to 
every point of the compass as the crank turns, so that 
the ball turns on its own axis as really as if it were 
whirling upon a pin fastened to the table. 

By virtue of its connection with the crank, U 

Fig. 69 

the ball has two distinct motions : (1) the 

motion of translation^ which carries its center in a circle around 

the axis of the shaft; (2) an additional motion of rotation'^ around Rotation 

a line drawn throug^h its center parallel to the shaft. The pin A ^^''^ ^^^^ 

^ ... carried by a 

(in the figure) and the hole in which it fits both rotate at the crank arm. 
same rate, so that the ball, while it turns on its " axis " (an 
imaginary line), does not turn 07i the pin^ nor the pin in the hole. 

203. Librations. — While in the "long run " the moon keeps The rotation 
the same face towards the earth, it is not so m the short run; of the moon 


there is no crank connection between the earth and moon, and of its orbital 
the moon in different parts of a single month does not keep revolution 


exactly the same lace towards the earth, but rotates with per- havino-the 
feet independence of her orbital motion. With reference to same period, 
the center of the earth the moon's face is continually oscillating 
slightly, and these oscillations constitute what are called libra- 
tions., of which we distinguish three, — viz., the libration in 
latitude., the libration in longitude., and the di^irfial libration. 

1 Rotation consists essentially in this : that a line connecting any two points, 
and not parallel to the axis of the rotating body, will sweep out a circle on the 
celestial sphere, if produced to it, 



of the 
equator to 
the plane of 
her orbit, 
libration in 

while orbital 
motion is 
libration in 

due to 
ment by the 
rotation of 
the earth. 




of periods of 
rotation and 
orbital revo- 
lution prob- 
ably to be 
by tidal 

(1) The libration in latitude is due to tlie fact that the moon's 
equator does not coincide with the plane of its orbit, but makes 
with it an angle of about 6^°. This inclination of the moon's 
equator causes its north pole at one time in the month to be 
tipped 6|-° towards the earth, while a fortnight later the south 
pole is similarly inclined to us ; just as the north and south 
poles of the earth are alternately presented to the sun, causing 
the seasons. 

(2) The libration in longitude depends on the fact that the 
moon's angular motion in its oval orbit is variable, while the 
motion of rotation is uniform, like that of any other undis- 
turbed body ; the two motions, therefore, do not keep pace 
exactly during the month, and we see alternately a few degrees 
around the eastern and western edge of the lunar globe. This 
libration amounts to about 7f°. 

(3) The diurnal libration. Again, when the moon is rising 
we look over its upper, which is then its western, edge, seeing a 
little more of that part of the moon than if we were observing 
it from the center of the earth ; and vice versa when it is setting. 
This constitutes the so-called diurnal libration, and amounts to 
about one degree. Strictly speaking, this diurnal libration is not 
a libration of the moon, but of the observer. T^he telescopic 
effect is the same, however, as that of a true libration. 

In addition to this there is also a very slight physical 
libration of the moon. It is probable that the diameter of 
the moon directed towards the earth is a little longer than the 
diameter at right angles, the difference being perhaps a few 
hundred feet ; and as the moon revolves around the earth this 
longest diameter oscillates slightly from side to side, changing 
its position apparently about 1|- miles on the moon's disk. 

The exact, long-run agreement between the moon's time of 
rotation and of her orbital revolution cannot be accidental. It 
has probably been caused by the action of the earth on some 



protuberance on the moon's surface, analogous to a tidal wave. 
If the moon were ever plastic, the earth's attraction must 
necessarily have been to produce a huge tidal bulge upon her 
surface, and the effect would have been ultimately to force an 
agreement between the lunar day and the sidereal month. The 
subject will be resumed later in connection with tidal evolution 
(Sec. 346). 

204. The Phases of the Moon. — Since the moon is an opaque 
body shining merely by reflected light, we can see only that 

Phases of 
the moon 
due to the 
fact that we 
see only a 
varying por- 
tion of her 

Fig. 70. — Explanation of the Moon's Phases 

hemisphere of her surface which happens to be illuminated, 
and of this hemisphere only that portion which happens to be 
turned towards the earth. When the moon is between the 



The termi- 
always a 

Direction of 
the horns of 
the crescent 
away from 
the sun. 

earth and the sun (at new moon) the dark side is then pre- 
sented directly towards ns, and the moon is entirely invisible. 
A week later, at the end of the first quarter, half of the illumi- 
nated hemisphere is visible, just as it is a week after the full. 
Between the new moon and the half-moon, during the first 
and last quarters of the lunation, we see less than half of 
the illuminated portion and then have the " crescent " phase. 
Between half-moon and the full moon, during the second and 
third quarters of the lunation, we see more than half of the 
moon's illuminated side and have then what is called the 
" gibbous " phase. 

Fig. 70 (in which the light is supposed to come from a 
point far above the circle Avhich represents the moon's orbit) 
shows how the phases are distributed through the month. 

205. The Terminator. — The line which separates the dark 
portion of the disk from the bright is called the terminator and 
is always a semi-ellipse, since it is a semicircle viewed obliquely. 
The illuminated portion of the moon's disk is, therefore, always 
a figure which is made up of a semicircle plus or minus a semi- 
ellipse, as shown in Fig. 71 A, At new 
or full moon, however, the semi-ellipse 
becomes a semicircle. It is sometimes 
incorrectly attempted to represent the 
crescent form by a construction like 
Fig. 71 B, in which a smaller circle has 
a portion cut out of it by an arc of a larger one. 

It is to be noticed also that ab, the line which joins the 
" cusps," or points of the crescent, is always perpendicular to 
a line drawn from the moon to the sun, so that the horns are 
always turned away from the sun. The precise position, there- 
fore, in which they will stand at any time is perfectly predict- 
able and has nothing whatever to do with the weather. Artists 
are sometimes careless in representing a crescent moon with its 
horns pointed downwards, which is impossible. 

Fig. 71 



206. Earth-Shine on the Moon. — Near the time of new 
moon the whole disk is easily visible, the portion on which 
sunlight does not fall being illuminated by a pale reddish light. 
This light is earth-shine^ the earth as seen from the moon being Earth-shi 
then nearly " full." on the moon. 

Seen from the moon, the earth would show all the phases that 
the moon does, the earth's phase being in every case exactly 
supplementary to that of the moon as seen by us at the time. 
Taking everything into account, the earth-shine by which the 
moon is illuminated near new moon is probably from fifteen to 
twenty times as strong as the light of the full moon. The 
ruddy color is due to the fact that the light sent to the moon 
from the earth has passed twice through our atmosphere and 
so has acquired the sunset tinge. 


207. The Moon's Atmosphere The moon's atmosphere, if No sensible 

any exists, is extremely rare, probably not producing- at the ^^^osphere 

'^ 'J ^ ^ r J ir & on the moon. 

moon's surface a barometric pressure to exceed ^V of an inch 
of mercury, or yi^- of the atmospheric pressure at the earth's 
surface. The evidence on this point is twofold. 

First, the telescopic appearance. The parts. of the moon near No haze, ail 
the edge of the disk, or "limb," which, if there were any atmos- ^^^^^^^ 

^ ^ ^ -^ perfectly 

phere, would be seen through its greatest possible depth, are black; no 
visible without the least distortion. There is no haze, and all ^^o^^^ or 


the shadows are perfectly black ; there is no evidence of clouds phenomena. 

or storms, or of anything like the ordinary phenomena of the 

terrestrial atmosphere. 

Second, the absence of refraction at the moon's limb, when the No sensible 

moon intervenes between us and any more distant object. At an ^® ^^^^^?^^ 

•^ '^ rays of light 

eclipse of the sun there is no distortion of the sun's limb where which pass 
the moon cuts it. When the moon " occults " a star there is close to the 

moon's limb. 

no distortion or discoloration of the star disk, but both the 



No water on 
the moon. 

How came 
the moon to 
lose her 
phere ? 

Partly, per- 
haps, by 
and chem- 
ical combi- 
nation in 

Perhaps by 

disappearance and reappearance are practically instantaneous. 
Moreover, an atmosphere of even slight density, quite insuffi- 
cient to produce any sensible distortion of the image, would 
notably diminish the time during which the star would be con- 
cealed behind the moon, since the refraction would bend the 
rays from the star around the edge of the moon so as to render 
it visible, both after it had really passed behind the limb and 
before it emerged from it. There are some rather doubtful 
indications of a very slight effect of this kind, corresponding 
to what would be produced by an atmosphere about yq^-q as 
dense as our own. 

208. Water on the Moon^s Surface Of course, if there is 

no atmosphere there can be no liquid water, since if there were 
it would immediately evaporate and form an atmosphere of 
vapor. It is not impossible, however, nor perhaps improbable, 
that solid water, ^.e., ice and snow, may exist on parts of the 
moon's surface at a temperature too low to liberate vapor 
eno.ugh to make an atmosphere observable from the earth. 

209. What has become of the Moon^s Air and Water? — If 
the moon ever formed a part of the same mass as the earth, 
she must once have had both air and water. There are a num- 
ber of possible, and more or less probable, hypotheses to account 
for their disappearance : (1) The air and water may have struck 
in, — partly absorbed by porous rocks and partly disposed of in 
cavities left by volcanic action ; partly also, perhaps, by chemi- 
cal combination as water of crystallization, and by simple occlu- 
sion. (2) The atmosphere may have flown away ; and this is 
perhaps the most probable hypothesis, though it is quite possible 
that this cause and the preceding may have cooperated. If the 
" kinetic " theory of gases is true, no body of small mass, not 
extremely cold, can permanently retain any considerable atmos- 
phere. A particle leaving the moon with a speed exceeding 
the " critical velocity " of 1^ miles a second would never 
return (Sec. 319). If she was ever warm, the molecules of her 


atmosphere must have been continually acquiring velocities 
greater than this, and deserting her one by one. (See Physics^ 
pp. 270, 271.) 

However it came about, it is quite certain that at present no 
substances that are gaseous or vaporous at low temperatures 
exist in any considerable quantity on the moon's surface, — at 
least, not on our side of it. 

210. The Moon's Light As to quality^ it is simply sunlight, Light of the 

showing a spectrum identical in every detail with that of light "^^^^^^ ^^ 
coming directly from the sun itself; and this may be noted identical 
incidentally as an evidence of the absence of a lunar atmosphere, ^}^^^ ^^^' 
which, if it existed in any quantity, would produce markings of 
its own in the spectrum. 

The brightness of full moonlight as compared with sunlight 
is estimated as about ^ "o oV"o "o • According to this, if the whole Light of full 
visible hemisphere were packed with full moons, we should ™°^^ ^^^^* 
receive from it about one-eighth part of the light of the sun. sunlight. 

Moonlight is not easy to measure, and different experimenters have 
found results for the ratio between the light of the full moon and sunlight 
ranging all the way from 300V00 (Bouguer) to -^-^-^-^-^-^ (Wollaston). The 
value now generally accepted is that determined by Zollner, viz., ^^^^^. 

The half-moon does not give, even approximately, half as Sudden 
much light as the full moon. Near the full the brightness ^^^^^^s® of 

Till 1- • brightness 

suddenly and greatly increases, probably because at any time near full 
except at the full moon the moon's visible surface is more or moon, 
less darkened by shadows. 

The average albedo., or reflecting power of the moon's surface, Moon re- 
ZoUner states as 0.174; ^.e., the moon's surface reflects a little fleets about 

one sixth of 

more than one-sixth part of the light that falls upon it. the light 

This corresponds to the reflecting power of a rather light- ^^i^hit 
colored sandstone. There are, however, great differences in 
the brightness of the different portions of the moon's surface. 
Some spots are nearly as white as snow or salt, and others as 
dark as slate. 




Heat re- 
ceived, from 
the full 
moon prob- 
ably about 
iBcW of that 
from the 


Mean tem- 
perature of 
the moon 
low, but 
range of 
very great. 

of opinion. 

211. Heat of the Moon. — For a long time it was impossible 
to detect the moon's heat by observation. Even when concen- 
trated by a large lens, it is too feeble to be shown by the most 
delicate thermometer. The first sensible evidence of it was 
obtained by Melloni in 1846, with the newly invented thermo- 
pile^ by a series of observations from the summit of Vesuvius. 

With modern apparatus it is easy enough to perceive the 
heat of lunar radiation, but the measurements are extremely 

A considerable percentage of the lunar heat seems to be heat 
simply reflected like light, while the rest, perhaps three quarters 
of the whole, is "obscure heat," z.e., heat which has first been 
absorbed by the moon's surface and then radiated, like the heat 
from a brick surface that has been warmed by sunshine. This 
is shown by the fact that a comparatively thin plate of glass 
cuts off some eighty-six per cent of the moon's heat. 

The total amount of heat radiated by the full moon to the 
earth is estimated by Lord Rosse at about one eighty -thousandth 
part of that sent us by the sun ; but this estimate is probably 
too high. Prof. C. C. Hutchins in 1888 found it y8 5Vo"o* 

212. Temperature of the Moon's Surface. — As to the tempera- 
ture of the moon's surface, it is difficult to affirm much with 
certainty. On the one hand, the lunar rocks are exposed to the 
sun's rays in a cloudless sky for fourteen days at a time, so 
that if they were protected by air like the rocks upon the earth 
they would certainly become intensely heated. During the 
long lunar night of fourteen days the temperature must inevi- 
tably fall appallingly low, perhaps 200° below zero. 

There have been great oscillations of opinion on this subject. 
Some years ago Lord Rosse inferred from his observations that 
the maximum temperature attained by the moon's surface was 
not much, if at all, below that of boiling water; but his own 
later investigations and those of Langley threw great doubt 
on this conclusion, rather indicating that the temperature never 


reaches that of melting ice. The latest observations, however — 
the elaborate work of Very in 1899 — corroborate Lord Rosse's 
earlier results and show almost conclusively that on the moon's 
equator at lunar noon the temperature rises very high, falling 
correspondingly low when night comes on. 

Lord Rosse has also found that during a total eclipse of the Sudden dis- 
moon her heat radiation practically vanishes and does not regain ^PP^arance 
its normal value until some hours after she has left the earth's heat when 
shadow. This seems to indicate that she loses heat nearly as ^n^niersed in 

... ' the earth's 

last as it IS received. shadow. 

213. Lunar Influences on the Earth. — The moon's attraction 
cooperates with that of the sun in producing the tides^ to be 
considered later. 

There are also certain distinctly ascertained disturbances of influences 
terrestrial magnetism connected with the approach and recession ^^ ^^^ ""°^,^ 

^ ^ •*• -^ _ on the earth : 

of the moon at perigee and apogee ; and this ends the chapter only tidal 
of ascertained lunar influences. action and a 

very slight 

The multitude of current beliefs as to the controlling influ- magnetic 

ence of the moon's phases and changes upon the weather and disturbance, 

the various conditions of life are simply superstitions, mostly Numerous 

unfounded or at least unverified. supersti- 

.. .,.„, , .^ ^^ c tions for 

It IS quite certain that ii the moon has any mnuence at all of which no 
the sort imagined it is extremely slight, so slight that it has not evidence can 
yet been demonstrated, though numerous investigations have 
been made expressly for the purpose of detecting it. It is not 
certain, for instance, whether it is warmer or not, or less cloudy 
or not, at the time of full moon. 

214. The Moon's Telescopic Appearance and Surface. — Even 
to the naked eye the moon is a beautiful object, diversified with 
markings which are associated with numerous popular myths. 
In a powerful telescope these markings mostly vanish and are 
replaced by a multitude of smaller details which make the The moon as 
moon, on the whole, the finest of all telescopic objects, — ^ telescopic 


especially so for instruments of a moderate size (say from 6 to 



Best time 
to look at 
the moon. 

The moon's 
surface very 
broken by 
a lew moun- 
tain ranges 
and numer- 
ous craters. 

of craters. 

10 inches in diameter), which generally give a more pleasing 
view of our satellite than instruments either much larger or 
much smaller. 

An instrument of this size, with magnifying powers between 
250 and 500, brings the moon optically within a distance rang- 
ing from 1000 to 500 miles ; and since an object half a mile in 
diameter on the moon subtends an angle of about 0''.43, it 
would be distinctly visible. A long line, or streak, even less 
than a quarter of a mile across can probably be seen. With 
larger telescopes the power can now and then be carried very 
much higher, and correspondingly smaller details made out, when 
the seeing is at its best, not otherwise. The right-hand illustration 
opposite gives an excellent idea of the moon's appearance with 
a moderate magnifying power of about 100. 

For most purposes the best time to look at the moon is when 
it is between six and ten days old. At the time of full moon 
few objects on the surface are well seen, as there are then no 
shadows to give relief. 

It is evident that while with the telescope we should be able 
to see such objects as lakes, rivers, forests, and great cities, if 
they existed on the moon, it would be hopeless to expect 
to distinguish any of the minor indications of life, such as 
buildings or roads. 

215. The Moon's Surface Structure. — The moon's surface for 
the most part is extremely broken. With us the mountains are 
mostly in long ranges, like the Andes and Himalayas. On the 
moon the ranges are few in number; but, on the other hand, 
the surface is pitted all over with great craters, which resemble 
very closely the volcanic craters on the earth's surface, though 
on an immensely greater scale. The largest terrestrial craters 
do not exceed 6 or 7 miles in diameter; many of those on the 
moon are 50 or 60 miles across, and some have a diameter of 
more than 100 miles, while smaller ones from 5 to 20 miles in 
diameter are counted by the hundred. 




















1 — ' 









2 hj 

«o O 

o o 

^. so. 

- p 

g p- 




The normal 
lunar crater. 


Fig. 72. — Normal Lunar Crater 

The normal lunar crater (Fig. 72) is nearly circular, sur- 
rounded by a ring of mountains which rise anywhere from 1000 

to 20000 feet above 
the surrounding 
country. The floor 
within the ring may 
be either above or 
below the outside 
level ; some craters 
are deep, and some 
filled nearly to the 
brim. In a few cases 
the surrounding 
mountain ring is en- 
tirely absent, and the crater is a mere hole in the plain. 
Frequently in the center 
of the crater there rises 
a group of peaks, which 
attain about the same ele- 
vation as the encircling 
ring, and these central 
peaks sometimes show 
holes or craters in their 
summits. Fig. 73 is from 
a drawing by Nasmyth of 
the crater Gassendi, which 
is on the southeast quad- 
rant of the moon's sur- 
face, and comes into view 
about three or four days 
before full moon. It is 
58 miles in diameter and 
about 8000 feet deep. 
Fig. 74 is also from one of Nasmyth' s drawings and is a fine 

Fig. 73. — Gassendi 



representation of Copernicus, a crater not quite so large or deep Copernicus, 
as Gassendi, but very interesting on account of the number of 
surrounding ridges and the manner in which the neighboring 
region is thickly sown with crate rlets and holes. It is on the 
terminator a day or two after the half -moon. 

In the enlarged photograph of a portion of the moon's XheopWius. 
surface on page 187 the great crater at the left is Theophilus, 
64 miles in diameter and 
nearly 19000 feet deep. 

On certain portions of 
the moon these craters 
stand very thickly ; older 
craters have been en- 
croached upon, or more 
or less completely obliter- 
ated, by the newer, so that 
the whole surface is a 
chaos of which the 
counterpart is hardly to 
be found on the earth, 
even in the roughest por- 
tions of the Alps. This 
is especially the case near 
the moon's south pole. It 
is noticeable, also, that as 
on the earth the youngest 

mountains are generally the highest, so on the moon the newer 
craters are generally deeper and more precipitous than the older. 

The height of a lunar mountain or depth of a crater can be 
measured with considerable accuracy by means of its shadow, 
or, in the case of a mountain, by the measured distance between 
its summit and the terminator at the time when the top first 
catches the light, looking like a star quite detached from the 
bright part of the moon, as seen in Fig. 73. 

Fig. 74. — Copernicus 


usually the 

ment of 
on the moon. 



craters are 
probably of 
origin, but 
the explana- 
tion is not 
free from 

No volcanic 
action at 
evident on 
the moon. 

Rills and 

Streaks, or 

216. The striking resemblance of these formations to terres- 
trial volcanic structures, like those exemplified by Vesuvius 
and others, makes it natural to assume that they had a similar 
origin. This, however, is not absolutely certain, for there are 
considerable difficulties in the way, especially in the case of 
what are called the great " Bulwark Plains." These are so 
extensive that a person standing in the center could not see 
even the summit of the surrounding ring at any point ; and yet 
there is no line of discrimination between them and the smaller 
craters — the series is continuous. Moreover, on the earth vol- 
canoes necessarily require the action of air and water, which do 
not at present exist on the moon. It is obvious, therefore, that 
if these lunar craters are the result of volcanic eruptions, they 
must be, so to speak, " fossil " formations, for it is quite certain 
that there is absolutely no evidence of present volcanic activity. 

217. Other Lunar Formations. - — The craters and mountains 
are not the only interesting formations on the moon's surface. 
There are many deep, narrow, crooked valleys that go by the 
name of " rills," some of which may once have been water- 
courses. Fig. 74 shows several of them. Then there are 
numerous straight " clefts," half a mile or so wide and of 
unknown depth, running in some cases several hundred miles, 
straight through mountain and valley, without any apparent 
regard for the accidents of the surface : they seem to be deep 
cracks in the crust of our satellite. Most curious of all are 
the light-colored streaks, or "rays," which radiate from certain 
of the craters, extending in some cases a distance of many hun- 
dred miles. These are usually from 5 to 10 miles wide and 
neither elevated nor depressed to any considerable extent with 
reference to the general surface. Like the clefts, they pass 
across valley and mountain, and sometimes through craters, 
without any change in width or color. They have been doubt- 
fully explained as a staining of the surface by vapors ascending 
from rifts too narrow to be visible. 


The most remarkable of these " ray systems " is the one con- 
nected with the great crater Tycho, not very far from the moon's Ray system 
south pole, well shown in the (nearly) full-moon photograph on ^^ Tycho. 
page 187. The rays are not very conspicuous until within a 
few days of full moon, but at that time they and the crater 
from which they diverge constitute by far the most striking 
feature of the whole lunar surface. 

218. Lunar Maps. — A number of maps of the moon have been con- Lunar maps, 
structed by different observers. The most extensive is that by Schmidt 
of Athens, on a scale 7 feet in diameter, published by the Prussian govern- 
ment in 1878. Of the smaller maps available for ordinary lunar observa- 
tion, perhaps the best is that given in Webb's Celestial Objects for Common 
Telescopes. Two new photographic, large-scale, lunar maps have lately 
been published from negatives made at the Lick and Paris observatories. 
Our maps of the visible part of the moon are on the whole as complete 
and accurate as our maps of the earth, taking into account the polar 
regions and the interior of the continents of Asia and Africa. 

219. Lunar Nomenclature. — The great plains upon the moon's surface Lunar 
were called by Galileo "oceans" or "seas" (tncma), for he supposed that nomeu- 
these grayish surfaces, which are visible to the naked eye and conspicuous 
in a small telescope, though not with a large one, were covered with water. 

The ten mountain ranges on the moon are mostly named after terrestrial 
mountains, as Caucasus, Alps, Apennines, though two or three bear the 
names of astronomers, like Leibnitz, Doerfel, etc. 

The conspicuous craters bear the names of eminent ancient and medi- 
eval astronomers and philosophers, as Plato, Archimedes, Tycho, Coperni- 
cus, Kepler, and Gassendi ; while hundreds of smaller and less conspicuous 
formations bear the names of more modern astronomers. 

This system of nomenclature seems to have originated with Riccioli, 
who in 1650 made the first map of the moon. 


220. Changes on the Moon. — It is certain that there are no Question of 
conspicuous chanofes : there are no such transformations as ^, ^^s^^ o^ 

-^ o ^ the moon s 

would be presented by the earth viewed telescopically, — no surface, 
clouds, no storms, no snow of winter, and no spread of vege- ^^^^® *^^* 

are obvious, 

tation in the spring. At the same time it is confidently ^^t some 
maintained by some observers that here and there alterations probable. 



do take place in the details of the lunar surface, while others 
as stoutly dispute it. 

The difficulty in settling the question arises from the great 
changes in the appearance of a lunar object under varying illu- 
mination. To insure certainty in such delicate observations, 

Fig. 75. — Map of the Moon 
Reduced from Neison 

Difficulty of comparisons must be made between the appearance of the object 
the problem. ^^ question, as seen at precisely/ the same phase of the moon, 
with telescopes (and eyes too) of equal power, and under sub- 
stantially the same conditions in other respects, such as the 
height of the moon above the horizon and the clearness and 



steadiness of the air. It is, of course, very difficult to secure such 
identity of conditions. (For an account of certain supposed 
changes, see Webb's Celestial Objects for Common Telescopes.) 

221. Fig. 75 is reduced from a skeleton map of the moon by Skeleton 
Neison and, though not large enough to exhibit much detail, "^^p ^^ *^^ 
will enable a student with a small telescope to identify the 
principal objects by the help of the key. 


kp:y to the principal objects indicated in fig. 75 

A. Mare Humorum. 

B. Mare Nectaris. 

C. Oceanus Procellarum. 

D. Mare Fecunditatis. 

E. Mare Tranquilitatis. 

F. Mare Crisium. 

G. Mare Serenitatis. 
H. Mare Imbrium. 

/. Sinus Iridum. 

K. Mare Nubiuni. 
L. Mare Frigoris. 
T. Leibnitz Mountains. 
U. Doerfel Mountains. 
V. Rook Mountains. 
W. D'Alembert Mountains. 
X. Apennines. 
Y. Caucasus. 
Z. Alps. 

1. Clavius. 

2. Schiller. 

3. Maginus. 

4. Schickard. 

5. Tycho. 

6. Walther. 

7. Purbach. 

8. Petavius. 

9. "The Railway." 

10. Arzachel. 

11. Gassendi. 

12. Catherina. 

13. Cyrillus. 

11. Alphonsus. 

15. Theophilus, 

16. Ptolemy. 

17. Langrenus. 

18. Hipparchus. 

19. Grimaldi. 

20. Flamsteed. 

21. Messier. 

22. Maskelyne. 
28. Triesnecker. 

24. Kepler. 

25. Copernicus. 

26. Stadius. 

27. Eratosthenes. 

28. Proclus. 
28'. Pliny. 

29. Aristarchus. 

30. Herodotus. 

31. Archimedes. 

32. Cleomedes. 

33. Aristillus. 

34. Eudoxus. 

35. Plato. 

36. Aristotle. 

37. Endymion. 

222. Lunar Photography. — It is probable that the question Lunar pho- 
of changes upon the moon's surface will in the end be authori- ^^g^'^P^y- 
tatively decided by means of photography. The earliest success 
in lunar photography was that of Bond of Cambridge, U.S., in 



in this 


1850, using the old daguerreotype process. This was followed 
by the work of De la Rue in England, and by Dr. Henry Draper 
and Mr. Lewis M. Rutherfurd in this country. Until very 
recently Mr. Rutherfurd's pictures have remained absolutely 
unrivaled; but since 1890 there has been a great advance. At 
various places, especially at Cambridge and at the Lick and 
Yerkes observatories in this country, and at Paris, most admi- 
rable photographs have been made, which bear enlargement well 
and show almost as much detail as can be seen with the telescope, 
— not quite, however. 

The half-tone engraving of the entire moon on page 187 is shghtly 
enlarged from a photograph made by Professor Hale at his Kenwood 
Observatory (Chicago) in 1892 with a 13^-inch photographic object-glass. 
The other covers a small portion of the moon's surface on a much larger 
scale, including the great crater Theophilus with its neighbors Cyrillus 
and Catherina. It is enlarged from a magnificent photograph made in 
1900 by Ritchey of the Yerkes Observatory with the non-photographic 
object-glass of the great 40-inch telescope, a yellowish color screen being- 
used in front of the sensitive plate to cut off the red, violet, and ultra-violet 
rays, according to the method introduced by Professor Hale. The original 
negative is certainly not surpassed by any thus far obtained with photo- 
graphic lenses or reflectors. 


Its Distance, Dimensions, Mass, and Density — Its Rotation and Equatorial Accel- 
eration — Metliods of studying its Surface — The Photosphere — Sun-Spots — 
Their Nature, Dimensions, Development, and Motions — Their Distribution and 
Periodicity — Sun-Spot Theories 

The sun is the nearest of the stars, — a hot self-luminous 
globe, enormous as compared with the earth and moon, though 
probably only of medium size compared with other stars ; but to 
the earth and the other planets which circle around it it is the 
most magnificent and important of all the heavenly bodies. Its The sun's 
attraction controls their motions, and its rays supply the energy '^^P^emacy. 
which maintains every form of activity upon their surfaces. 

223. The Distance of the Sun ; the Astronomical Unit. — The 
problem of finding accurately the suyi's distance is one of the 
most important and difficult presented by astronomy, — impor- importance 
tant because this distance, i.e., the radius of the earth's orbit, is ^^^ ^^^' 

culty of 

the fundamental Astronomical Unit to which- all measurements determining 
of celestial distance are referred ; difficult because the measure- *^® distance 

.. -, -.. ^ . of the sun, 

ments which determine it are so delicate that any minute error ^l^Q funda- 
of observation is enormously magnified in the result. mental 

Without a knowledge of the sun's distance we cannot form .^^^ ^^^^ 
any idea of its real dimensions, mass, and density, and the 
tremendous scale of solar phenomena. 

We have already given one method for finding this distance, 
depending upon the experimental determination of the velocity Already 
of light, combined with the observed constant of aberration, and determmed 
we postpone until later the consideration of the methods by tion of light, 
which we measure the sun's parallax (Sec. 79) and so determine 




Its distance 
92 900000 





his distance in terms of the radius of the earth. From the 
combination of all the material now available the snn's mean 
distance comes out very closely 92 900000 miles (149 500000 
kilometers), the horizontal parallax being 8'^80 ± 0'^02. 

The distance is still uncertain by perhaps 100000 miles, and 
because of the eccentricity of the earth's orbit it is variable to 
the extent of about 3 000000 miles, being the least on January 1 
and greatest early in July. 

The orbital velocity of the earth, found by dividing the cir- 
cumference of the orbit by the number of seconds in a year, 
is 18|- miles a second, as already determined by aberration 
(Sec. 173). (Compare this velocity with that of a cannon-shot 
— seldom exceeding 2500 feet per second.) 

Perhaps one of the simplest illustrations of the distance of the sun is 
that such a shot would require over six years to reach the sun, traveling 
without change of speed. A railroad train running at 60 miles an hour, 
without stop or slackening, would require 175 years, and the fare one way, 
at two cents a mile, would be %1 860000. A bicyclist traveling 100 miles 
a day would be nearly 2550 years in making the journey, and if he had 
started from the sun in the year a.d. 1, he would by this time have covered 
only about three quarters of the distance. Light makes the journey in 
499 seconds. 

The sun's 
109.5 times 
that of the 

Illustration : 
radius of 
the sun com- 
pared with 
the distance 
of the moon. 

224. Dimensions of the Sun. — The sun's mean apparent 
diameter is 32' 4'' ± 2". Since at the distance of the sun one 
second equals 450.36 miles (92 900000^206264.8), its real 
diameter is 866500 miles, or 109^ times that of the earth. It 
is quite possible that this diameter is variable to the extent 
of a few hundred miles, since the sun is not solid. 

If we suppose the sun to be hollowed out, and the earth 
placed at the center, the sun's surface would be 433000 miles 
away. Now, since the distance of the moon from the earth is 
about 239000 miles, she would be only a little more than half- 
way out from the earth to the inner surface of the hollow globe, 
which would thus form a very good sky background for the 



study of the lunar motions. Fig. 76 illustrates the size of the 
sun, and of such objects upon it as the sun-spots and ''promi- 
nences," compared with the size of the earth and the moon's 

If we represent the sun by a globe 2 feet in diameter, the earth on 
the same scale would be 0.22 of an inch in diameter, the size of a very 
small pea, at a distance from the sun of just about 220 feet ; and the 
nearest star, still on the same scale, ivould he 8000 miles away at the antipodes. 

Fig. 76. — Dimensions of the Sun compared with the Moon's Orbit 

Since the surfaces of globes are proportional to the squares 
of their radii, the surface of the sun exceeds that of the earth 
in the ratio of 109. 5^ : 1 ; i.e.^ the area of its surface is about 
12000 times the surface of the earth. 

The volumes of spheres are proportional to the cubes of their 
radii. Hence, the sun's volume (or hulk) is 109.5^, or 1 300000^ 
times that of the earth, 

225. The Sun^s Mass. — This is about 333000 times that of 
the earth. For our purpose the most convenient way of reaching 

Surface area 
of the sun 
12000 times 
as great as 
that of the 
earth. Bulk 
1 300000 
times as 
great as that 
of the eartli. 



Mass of the 
sun 333000 
times that 
of the earth. 

How deter- 

orbit departs 
from a 
straight line 
only a little 
more than 
one ninth of 
an inch in 
18.5 miles. 

this result (for another method, see Sec. 380) is by comparing 
g^^ the earth's attraction for bodies at her surface, with the sun's 
attraction for the earth as measured by our orbital motion. 
Call this attraction /i, and let B be the radius of the earth's 
orbit, r the radius of the earth, S the mass of the sun, and E 
that of the earth. Then, from the law of gravitation (Sec. 146), 

we have 




But, from the law of central force, f ='—-< in which V is 


181" miles a second and R 92 900000 miles. Reducing R and V 
to inches, and making the computation, we find/= 0.2332 inches, 

and since g (corrected for centrifugal force) is 386.8 inches, - = 

1 K. ^ 92 900000 ^^.^^ ,, , /. , 

-— —-. Also, - = = 23467, the square of which 

1658. 7 r 0958.8 aQaf\f\f\ 

is 550 686000. Finally, therefore, ^ = ^^ X ^ ^^ = 

332400 JS. But the last three figures are uncertain. 

226. The Curvature of the Earth^s Orbit and Total Force of 
Sun's Attraction. — The distance which the earth would fall 
towards the sun in a second if its orbital motion were arrested 
is ^ /, or 0.116 inches, just as ^ g, IQ^j feet, is the distance 
a body falls towards the earth in the first second; and this 
0.116 inches is the amount by which the earth deviates from a 
tangent to its orbit in a second. In other words, the earth in 
traveling 18.5 miles is deflected towards the sun but a little 
more than one ninth of an inch. 

1 Since the attractions of the sun and earth are here measured by the acceler- 

S + E E + m 

ations f and g, the proportion would strictly he f : g = 
m is the small body by the fall of which g is determined. 


i?2 r2 

But E and m are so 

small as compared with S and E, respectively, that they may be omitted without 

sensible error, as in the proportion given. 

THE SUN 199 

It would seem that a feeble force only would be needed to 
produce so slight a deviation from a straight line. But since 
the sun's attraction is j-^-^q g^ a mass of 1659 pounds on the 
earth is attracted towards the sun with a force of about one 

It follows, therefore, that the total attraction between the Total attrac- 
earth and sun amounts to the amazing pull of 3 600000 millions tiont>etween 

T , , . . ^ . sun and 

of millions of to7is (y 0^5-9- of the earth s mass, which is 6 x 10^^ earth equal 

tons). This would be the breaking strain of a steel rod more ^^ ^^^ 

than 3000 miles in diameter, — a force inexplicably exerted strain of a 

through, or transmitted by, apparently empty space, in which steel rod 

,1 n , -,1 , .11 . , 3000 miles in 

the planets move without sensible resistance. diameter 

227. Superficial Gravity, or Gravity at the Surface of the 
Sun. — This is found by dividing the sun's mass by the square 

of its radius (both compared with the earth), z.e., ^ , which Gravity on 

(10 9.0 2) sun's sur- 

gives 27.6. A mass of ten pounds would weigh 276 pounds face nearly 
on the sun, and a person who weighs 150 pounds here would " ^^ ^^ ^^ 
weigh over tiuo tons there. Locomotion would be impossible, the earth. 
A body would fall 444 feet in the first second, and a pendu- 
lum which here vibrates in a second would vibrate in less than 
one fifth of a second there. But (putting temperature out of 
consideration) a watch would go no faster there than here, since 
neither the inertia of the balance-wheel nor the elasticity of 
the spring would be affected by the increased gravity. 

228. The Sun^s Density. — Its mean density as compared Mean 
with that of the earth may be found by simply dividing^ its mass <^®"sity of 

. -^ . , , . the sun only 

by its volume (both as compared with the earth) ; z.e., the sun s about one 
density equals 332000 - 1 300000 = 0.255, — a little more fourth of 

, » . 111. the earth's 

than one quarter 01 the earth s density. density, or 

1.4 times 
1 This does not imply, and it is not true, that when the sun is overhead a 166- that of 
pound man w^eighs one tenth of a pound less (by a spring-balance) than w^hen water, 
the sun is rising. (Why not ?) The difference is really only about 20 oooooo 
of his weight (Sec. 333). 



Its signifi- 
cance as 
the sun's 

Axial rota- 
tion of the 
sun. Aver- 
age appar- 
ent or 
27.25 days. 

Equation for 
the sidereal 
period from 
the synodic. 

To get its specific gravity — its density compared with water — 
we must multiply this by the earth's mean specific gravity, 
5.53, which gives 1.41. That is, the sun^s mean density is less 
than one and one-half times that of water. 

This is a most remarkable and significant fact, considering the 
sun's tremendous force of gravity and that a considerable portion 
of its mass is composed of metals, as proved by the spectroscope. 
The obvious, and only possible, explanation is that the tempera- 
ture of the sun is such that its materials are almost wholly in 

the condition of vapor, — not 
solid or even liquid. 

229. The Sun»s Rotation. — 
This is made evident by the 
behavior of the dark sun-spots 
which cross the sun's disk from 
east to west. The times in 
which they make their circuits 
differ slightly, but the aver- 
age, as seen from the earth, 
is 27.25 days. This, however, 
is not the true or sidereal time 

Fig. 77. -Synodic and Sidereal Revolu- ^f ^-^^ ^^^^^^ rotation, but the 
tion of the Sun 

synodic, as is evident from 
Fig. 77. Suppose that an observer on the earth at E sees a spot 
on the center of the sun's disk at S; while the sun rotates the 
earth will also move forward in its orbit, and when he next sees 
the spot on the center of the disk he will be at U\ the spot 
having gone around the whole circumference plus the arc SS'. 
The equation by which the true period is deduced from the 
synodic is the same as in the case of the moon, viz., 

1_ J__2 

T being the true sidereal period of the sun's rotation, E the 
length of the year, and S the observed synodic rotation. This 


gives T= 25.35 days. In a year the number of sidereal revolu- Sidereal 
tions exceeds that of the synodic by exactly one. period about 

25.3 days 

Different observers, however, get slightly different results, 
because the spots are not fixed in their positions on the sun's 
surface. Carrington finds 25.38 days and Spoerer 25.23 days. 

The paths of the spots across the sun's disk are usually 
more or less oval, showing that the sun's axis is not perpen- inclination 
dicular to the ecliptic, but so inclined that the north pole is ^^ ^^^'^ 

1 T 1 T >-rn 11 • • equator to 

tipped a little more than 7 towards the position which the the ecliptic 
earth occupies on September 7. The inclination of the sun's about 7°. 
equator to the plane of the terrestrial equator is about 26° 15'; 
but different investigators get slightly different values. 

The position of the point in the sky towards which the sun's Position of 
axis is directed is *^® ^^^'^ 

J / ^4~^ Pol6 in the 

in right ascension /^^\ /^^\ /^^\ X"^^ heavens. 


Fig. 78. — Path of Sun-Spots across the Sun's Disk 

19% declination 

+ 63° 48', very 

nearly half-way 

be t we en the 

bright star a Lyrse 

and the pole-star. Twice a year, when the earth is in the plane Dates when 

of the sun's equator, the sun-spot paths become strais^ht, — on earth passes 

, ' c, -,. rr the plane of 

June 3 and December 5. (See Fig. 78.) the sun's 

230. The Equatorial Acceleration. — It was noticed quite early equator, 
that different spots give different results for the period of rota- 
tion, but the researches of Carrington about 1860 first brought 
out the fact that the differences are systematic, and that at the 
solar equator the time of rotation is shorter than on either side of 
it. For spots near the sun's equator it is about twenty-five 
days ; in solar latitude 30°, twenty-six and one-half days ; and 
in solar latitude 40°, twenty-seven days. In latitude 45° it is The sun's 
fully two days long-er than at the equator; but we are unable equatorial 

'J 'J o ± ' acceleration. 

to GBiTTy the observations to higher latitudes because spots 
almost never appear beyond the parallels of 45°. 



FormulfB for Various f ormulge have been proposed to represent the motion ; that of 
sun's rate of Faye, which agrees with the observations as well as any, is A' = 862' — 
186' sin ^/, X being the daily motion. Spoerer's formula, as modified by 
Wolfer, is X = 8° 55' + 5°. 80 cos Z. This looks very different from Faye's, 
but gives very nearly the same results for the regions in which spots are 
observable. Kone of the formulae proposed rest on any sound theoretical 

rotation in 
solar lati- 

Sun's sur- 
face not 

Cause of 



not yet 



Probably a 


from past 


ments for 
study of 
sun's sur- 

and screen. 

Fig. 79. — Telescope and Screen 

Clearly the sun's visible surface is not solid, but permits 
motions and currents like those of our air and oceans. It 
might be argued that the spots misrepresent the sun's real rota- 
tion, not being fixed upon its surface, floating like our clouds. 
The faculae, however, give substantially the same result, and so 

do Duner's spectroscopic observa- 
tions of the shift of lines in the 
spectrum at the eastern and western 
limbs. (See Sec. 254.) 

PossiHy this equatorial acceler- 
ation may be, in some way, an effect 
of the tremendous outpour of heat 
from the solar surface, as Emden 
of Munich attempts to show in a 
paper just published. Other recent investigators, however, 
have reached the conclusion that it cannot be explained by 
causes now acting, but is a lingering survival from the sun's 
past history^ and destined ultimately to disappear. 

231. Arrangements for the Study of the Sun's Surface — The 
heat and light of the sun are so intense that we cannot look 
directly at it with a telescope. A very convenient method of 
exhibiting the sun to a number of persons at once is simply to 
attach to a telescope a small frame carrying a screen of white 
paper at a distance of a foot or more from the eyepiece, as 
shown in Fig. 79. A screen should also be used at the object 
end, as shown in the figure, in order to shade the paper upon 
which the image is formed. When the focus is properly 



pieces and 

adjusted a distinct image appears, which shows the sun's princi- 
pal features very fairly ; indeed, with proper precautions, almost 
as well as the most elaborate apparatus. Still, it is generally 
more satisfactory to look at the sun directly with a suitable 

With a small telescope, not more than 2^ or 3 inches in Solar eye- 
diameter, a simple shade glass is often used between the eye- 
piece and the eye ; but the dark glass soon becomes very hot scopes. 
and is apt to crack. With larger instruments it is necessary 
to use eyepieces specially designed for the purpose, and known 
as solar eyepieces^ or helioscopes^ which reject most of the light 
coming from the object-glass and 
permit only a small fraction of it to 
enter the eye. 

The simplest of them, and a very good 
one, is known as Herschel's, in which the 
sun's rays are reflected at right angles by 
a plane of unsilvered glass. The reflector 
is made wedge-shaped, as shown in Fig. 80, 
in order that the reflection from the back 
surface may not interfere w^ith the image. 
Most of the light passes through the glass 
and out through the open end of the eye- 
piece, but the reflected light is still too 

intense for the unprotected eye. Only a thin shade glass is required, how- 
ever, which does not become very much heated. A more elaborate polar- 
izing helioscope, figured in the General Astronomy, is still better. 

It is not a good plan to cap the object-glass in order to reduce 
the light. To cut down the aperture is to sacrifice the defini- 
tion of delicate details (Sec. 46). 

232. The Heliograph. — In the study of the sun's surface, 
photography is for some purposes very advantageous and much Soiarpho- 
used. The instrument (called a heliograpJi) must, however, have ^J^s^^^p^^'J 
lenses specially constructed for photography, since a visual graph, 
object-glass would be nearly worthless for the purpose unless 

Fig. 80. — Herschel Eyepiece 



and disad- 
of photog- 

possibly the use of a color screen might make it available 
(Sec. 222). Arrangements must be made also to secure an 
extremely rapid exposure, and it is best to use special slow 
plates. The disk of the sun on the negatives is usually from 
2 to 10 inches in diameter, but photographs of small portions 
of the solar surface are often made on a very much larger scale. 


Fig. 81. — Greenwich Photograph of Sun-Spot, Sept. 10, 1898 

Fig. 81 is reduced from a 9-inch photograph made with one of 
the heliographs at Greenwich, on Sept. 10, 1898. 

Photographs have the great advantage of freedom from pre- 
possession on the part of the observer, and in an instant of 
time secure a picture of the whole surface of the sun such as 
would require hours of labor for a skilful draftsman. On the 



other hand, they take no advantage of the instants of fine seeing, 
but offer merely, as some one puts it, "a brutal copy" of what- 
ever happened to appear when the plate was exposed, affected by 
all the momentary distortions due to atmospheric disturbance. 

233. The Photosphere. — The sun's visible surface is called xhephoto- 
the Pliotosphere^ i.e., the " li^ht sphere." When studied with sphere: its 

-^ o i. appearance. 

Fig. 82. — Nodules and Granules on the Sun's Surface 
After Langley 

a telescope under favorable conditions, and a rather low power, 
it appears not smoothly bright, but mottled, looking much like 
rough drawing-paper. It is considerably darker at the edge than 
in the center, the difference between the center and limb being 
especially conspicuous in photographs, as in Fig. 81. With a 
high power and the best atmospheric conditions, the surface is 



rice grains, 

The photo- 
sphere prob- 
ably a stratum 
of incandes- 
cent clouds, 
acting like 
a Welsbach 

shown to be made up, as seen in Fig. 82, of a comparatively 
darkish background sprinkled oyer with grains, or " nodules," 
as Herschel calls them, of something much more brilliant, — 
" like snowfiakes on gray cloth," according to Langley. These 
nodules, or " rice grains," are from 400 to 600 miles across, 
and when the seeing is best they themselves break up into 
" granules " still more minute. Generally the nodules are 
about as broad as they are long, though irregular, but here 
and there, especially in the neighborhood of the spots, they are 
drawn out into long streaks, and then are called "filaments," 

"willow leaves," or 
"thatch straws." 

Certain bright 
streaks and patches 
called Faculoe are also 
usually visible here 
and there upon the 
sun's surface, and 
though not very ob- 
vious near the center 
of the disk, they be- 
come conspicuous 
near the limb, espe- 
cially in the neighborhood of spots, as shown in Fig. 83. Prob- 
ably they are of the same nature as the rest of the photosphere, 
only elevated above the general level and intensified in bright- 
ness because less affected by the absorption of the overlying 

The photosphere is probably^ according to the view now 
generally accepted, a sheet of clouds floating in a less luminous 
atmosphere^ just as the clouds formed by the condensation of 
water vapor float in our air. It is intensely brilliant, for the 
same reason that the mantle of a Welsbach burner outshines 
the gas flame which heats it; the radiating power of the solid 

Fig. 83. — Spots and Faculse 
. After De la Rue 



and liquid particles which compose the clouds is extremely high 
as compared with that of the gases in which they float. (See 
also Sec. 278.) 

234. Sun-Spots. — Sun-spots, whenever visible, are the most 
conspicuous and interesting objects upon the solar surface. 
The appearance of a normal sun-spot (Fig. 84), fully formed 
and not yet beginning to break up, is that of a dark central 
umhra^ with a fringing penumbra composed of converging fila- 
ments. The umbra itself 

-N A ^< I U 

I (s,^^'^^. 

is not uniformly dark 
throughout, but is over- 
laid with filmy clouds, 
which usually require a 
good helioscope to make 
them visible, but some- 
times, though rather 
inf re qu ently , are c o n- 
spicuous, as in the figure. 
Usually, also, within the 
umbra there are a num- 
ber of round and very 
black spots, sometimes 
called " nucleoli," but 
often referred to as 
"Dawes' holes," after the 

name of their first discoverer. Even the darkest portions of the 
sun-spot, however, are dark only by contrast. Photometric 
observations show that the umbra gives about one per cent as 
much light as a corresponding area of the photosphere, so that 
the blackest portion of a sun-spot is really more brilliant than a 
calcium light (Sec. 265). 

Very few spots are strictly normal. They are often gathered 
in groups within a common penumbra, which is partly covered 
with brilliant "bridges" extending across from the outside 

The normal 


nucleoli, etc. 

Fig. 84. — Normal Sun-Spot 
After Secchi 

part of sun- 
spot as 
bright as a 



ties in 

believed to 
be usually 
in the photo- 

photosphere. Frequently the umbra is not in the center of 
the penumbra, or has a penumbra on one side only; and the 
penumbral filaments, instead of converging regularly towards 
the nucleus, are often distorted in every conceivable way. 
Fig. 85 is enlarged from a Greenwich photograph of the 
spot of September, 1898. 

235. Nature of Sun-Spots Until very recently sun-spots 

have been believed to be cavities in the photosphere filled with 
gases and vapors, cooler, and therefore darker, than the sur- 
rounding region. This theory is founded on the fact that 

Fig. 85. — Group of Spots from a Greenwich Photograph, Sept. 11, 1898 

many spots as they cross the sun's disk behave as shown in 
P^'ig. 86. Near the limb they look just as they would if they 
were saucer-shaped hollows, with sloping sides colored gray 
and the bottom black. 

This theory has, however, of late been seriously called in 
question ; many spots, possibly a majority, as shown by photo- 
graphs and drawings, fail to present the appearances described. 
But the principal objection lies in the behavior of spots in 



respect to their heat radiation. Near the center of the disk Evidence 
the thermopile shows that, as they are darker, so also they *^^*^^ 

^ ^ '^ some cases 

emit less heat than the photosphere around them ; but near the they are 
" limb " (^.e., the edge of the sun's disk) the difference becomes elevated 

, . IP M above it. 

less and ni some cases is even reversed, a tact most easily 
explained by supposing the spot to be high above the photo- 

On the whole, it now seems most probable that different spots 
lie at very different levels, some low down, forming hollows in 
the photosphere, but others at a considerable elevation. 

The penumbra is usually composed of ''thatch straws," or The 
long-drawn-out filaments of photospheric cloud, and these, as penumbra. 

... :U t -- * - 

^ ^ ■"''■fss-l 




Fig. 86. — Sun-Spots as Cavities 

has been said, converge in a general way towards the center of 
the spot, though not infrequently more or less spiral in their 

At its inner edge the penumbra, from the convergence of 
these filaments, is usually brighter than at the outer. The Terminal 
inner ends of the filaments are ordinarily club-formed ; but ^^"^^ ^f , 

•^ _ penumbral 

sometimes they are drawn out into fine points, which seem to filaments. 
curve downward into the umbra, like the rushes over a pool of 
water. The outer edge of the penumbra is usually pretty 
sharply bounded, and there the penumbra is darkest. In 
the neighborhood of the spot the surrounding photosphere is 
usually much disturbed and elevated into facuko, which ordi- 
narily appear before the spot is formed and continue after it 



Size of 

visible to 
the naked 

Duration of 
and faculae. 

Tendency of 
sun-spots to 
recur at 
points on 
sun's surface 
where spots 
have dis- 

Initial stage 
in life of a 

account of 
the later 

236. Dimensions of Sun-Spots. — The diameter of the umbra 
of a sun-spot varies all the way from 500 miles, in the case of 
a very small one, to 40000 or 50000 miles, in the case of the 
largest. The penumbra surrounding a group of spots is some- 
times 150000 miles across, though that is exceptional. Not 
infrequently sun-spots are large enough to be visible vi^ith the 
naked eye and can actually be thus seen at sunset or through a 
fog or by the help of a colored glass. 

The Chinese have many records of such objects, but their real 
discovery dates from 1610, as an immediate consequence of 
Galileo's invention of the telescope. Fabricius and Scheiner, 
however, share the honor with him as being independent 

237. Duration, Development, and Changes of Spots. — The 
duration of sun-spots is very variable ; but they are always, 
astronomically speaking, short-lived phenomena, sometimes last- 
ing for a few days only, though more usually, if of any size, for 
two or three months. In a single recorded instance (1840-41) 
a spot persisted for eighteen months. 

The faculse in the surrounding region generally endure much 
longer than the spots, and not infrequently a new group of 
spots breaks out in the same region where one has disappeared 
some time before, — as if the local disturbance which caused the 
spots and faculce still continued deep below the surface. 

The development of a spot or spot group usually begins, 
according to Secchi, with the formation of faculse interspersed 
with small dark points, or " pores." These pores grow rapidly 
larger, coalesce, and the neighboring " granules " of the photo- 
sphere are transformed into the filaments of the penumbra, 
converging towards the umbra. Ordinarily this process takes 
several days, but sometimes only a few hours. 

According to Cortie, the irregular group of scattered incipient 
spots soon passes into a second stage, stretching out east and 
west with two predominant spots, one a leader, the other a 

THE SVN 211 

rear-guard of the flock. The preceding one (in the direction of 
the sun's rotation) is usually more compact and regular, though 
the other is sometimes the larger. The leader apparently 
pushes forward upon the photosphere and so increases the 
length of the train of " spotlets " between the two principals. 
Then a third stage follows, as well shown in Fig. 85, Sec. 234. 
After a time these small spots generally disappear, usually 
followed pretty soon by the larger spot in the rear, leaving the 
leader to settle down into a well-formed "normal" spot, which 
may endure without much change for weeks or months ; not 
infrequently, however, the leader disappears with the rest. 
Frequently a large spot divides into several, separated by bril- 
liant bridges, and the " segments " fly apart with a speed of Segmenta- 
sometimes a thousand miles an hour. An active spot is an tion of spots 
extremely interesting telescopic object ; not infrequently a 
single day works a complete transformation. 

When a large spot vanishes it is most usually by the rapid 
encroachment of the surrounding atmosphere, which seems, as 
Secchi expresses it, to " tumble pell-mell into the cavity," if it 
be one, forming 2if acuta to replace the spot. 

238. Proper Motions of the Spots. — Spots within 15° of the 
equator usually drift slightly towards it, while those in higher Drift of sun- 
latitudes drift from it ; but the drift in latitude is seldom rapid, ^^^^^ ^^ 

^ ' latitude. 

and exceptions to the rule are numerous. 

Active spots as a rule drift pretty steadily forward in the 
direction of the sun's rotation. The quiet ones move slowly. Eastward 
if at all. Within and close around the spot the motion with ^^^^^ °^ 

^ active spots, 

reference to the spot is usually inward and downward, so far as 
it can be observed. Occasionally fragments of the penumbral Vertical 
filaments break off, move towards the center of the spot, and ^° ^^'j.(j 
disappear as if swallowed up by a vortex (but there are other in umbra, 
possible explanations of their vanishing, such as dissolving' into ^P^^^ . ^^ 

, , , ° . region ]ust 

invisible vapor). Sometimes, but rarely, the downward motion outside the 
in the umbra of a spot is swift enough to be detected by the penumbra. 



displacement of lines in the spectrum (Sec. 254). On the other 
hand, around the outer edges of the penumbra there is often a 
vigorous boiling up from below, evidenced by the eruption of 
prominences (Sec. 260) and by spectroscopic phenomena within 
the spot itself. Cyclonic action is often observed ; sometimes 
there are two or more " whirlpools " within the same spot, not 
infrequently rotating in opposite directions. 

239. Distribution of the Spots. — For the most part the spots 

are confined to two belts between 5° and 40° of north and south 

Distribution latitude (Fig. 87). A few appear near the equator at the time 

of spots m ^£ ^^g sun-spot maximum, and practically none beyond the 

each side forty-fifth degree, though in somewhat higher latitudes what 




of sun's 




December 6 *'' 

June 5'* 

3Iarch 6<ft 
Fig. 87. — Spot Belts and Paths 

September 5«* 

Trouvelot calls " veiled spots " sometimes appear, looking like 
dark masses floating a little below the surface of the photo- 
sphere and only dimly seen through the overlying cloud. 

Generally the numbers are about equal in the two hemispheres, but 
sometimes there is a marked difference for years. From 1672 to 1704 
not a single spot was observed on the northern hemisphere, and the break- 
ing out of a few in 1705 occasioned great surprise and was reported to 
the French Academy as an anomaly. No reason for such a one-sided 
inactivity has thus far been discovered. 

Periodicity 240. Sun-Spot Petlodicity. — The number of spots varies 
of sun- of-reatly in different years and shows an approximately res^ular 

spots: the ° ^ ^, , -^ ^^ "^ f 

eleven-year periodicity of about eleven years. The fact was first discov- 
cycie. gpg(j \yj Schwabe of Dessau, in 1843, as the result of his 



systematic watching of sun-spots for nearly twenty years, and 
has since been abundantly confirmed. 

Wolf of Zurich, who died in 1893, has collected all the obser- 
vations available and summarized them in the diagram of which 
Fig. 88 is the reproduction. Fig. 89 continues it to date. The 

■ 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 4 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 ' 

^5 -iV 

i ^ t^ 

\ \ TFn7"7f"S ,^TTJI:^PCT ]>^r7 7irR7^7? ^ 

10ft -- ^ li/j-— ^ — — 

-r- ^ r. . 4^ t 

A^^n^i \ ^ -j7^k-tq issp 

\ \ "^ '■ ^ ir'arsj, 

^f, ^ X U^ \ t V t 

^ ^ 4 ^^ Tl N -, X -,2^^ A- 

^ t aSl 5 ^ t \ t ^\ U- 

t I "\^zt 4: \ ^ ^ Jl 5^--- J^ 

\ 1 s2^ ^ t \ ^S" , ''\ \ it 

ir ^^ ^^ \^'^ ^-N. .yf-mrrdrm --:t-^-> 

" i7.S(i 790 ■800 :.810 !820 


t^ ^ 5 

- f ^^ i ^^^ 

-/^ r ^ T 4^ ^ \ 

^Eo/'-ig vt ^ ^"4 -r -Hi^ ?^^ -,^ t~^ 

1 \ i _U^-F'ra:/i;e // \\ . // \ >rcgxt \\ \ 

pi.^J i X -tL 4 v^ 4A -i A- 

L^T-^ IS II "^ t -CR ^ S ^ 

-^0 -7 % t \ t ''^\, dt ^X ^ t^ - 1 

^^4^ ^ '^ 7 ?- + ^-^ 5 5r J 

f V -t % t ± S^ 5 \t f S -," 

-li ^ G-a« -^^/'' ^ /- '^^L 2^ i 

7 ^ 7^ ^-i ^^ ^^^/ v^ S f 

00 ^^ _ ^^ ^^ -^==«^ 

S3( 184(1 SfMi 8H(i 187( 1880 

Fig. 88. — Wolf's Sun-Spot Numbers 

last maximum occurred in 1894; 
the last minimum near the begin- 
ning of 1901. 

During the maximum, the sur- 
face of the sun is never free 
from spots ; sometimes a hundred 
are visible at once. During the 
minimum, weeks, and months 



inn '/i^' 

100 ^A, 

^^^ t % 1 

1 % i^ ^ 7 

^ \ i I ^ 

50 ^ % ^ 

% ^ 1 \ t 

t % i j^ t 

% t ^v-^ - 

^^.-v ^ / 

-'^ '^-^■ 

18 80 1890 19 00 

Fig. 89. — Continuation of Fig. 88 

even, pass without a single one. The rise from minimum to 
maximum is much more rapid than the fall that follows, and 
evidently from the diagram the maxima are not of equal inten- 
sity, nor are their intervals equal. Dr. J. S. Lockyer (son of 
Sir Norman), from a recent investigation of all data (including 



not yet 

law of 

theories : 
none fully 

magnetic as well as strictly solar) between 1833 and 1901, finds 
that this variation appears to be itself periodical, with a period 
of about thirty-five years. But the time covered by the material 
is hardly sufficient to warrant a sure conclusion. 

Many attempts have been made to connect these phenomena 
in some way with planetary action, but so far without success, 
and the general impression has lately been that it is probably 
due to causes within the sun itself or its atmosphere — a sort of 
geyser-like action — rather than to anything external. 

But a recent thorough investigation by Professor Newcomb 
puts rather a different phase upon the matter. He finds a 
regular period of 11.13 years (4.62 4- 6.51) as a uniform cycle 
underlying the periodic variations of sun-spot activity ; just as 
the regular period of 365^ days underlies the more or less vari- 
able seasons. He adds, "Whether the cause of this cycle is 
to be sought in something external to the sun, or within it, . . . 
we have at present no way of deciding." 

241. Spoerer's Law of Sun-Spot Latitudes. — Speaking broadly, 
the disturbance which produces the spots of a given period first 
appears in two belts, about 30° north and south of the sun's 
equator. These seats of disturbance then move gradually 
towards the equator, and the spot maximum occurs when their 
latitude is about 16°, while the disturbance dies out at a lati- 
tude of from 5° to 10°, about thirteen or fourteen years after 
its first outbreak. Two or three years before this disappear- 
ance, however, two new zones of disturbance show themselves 
in latitude 30° to 35°. Thus, at the spot minimum there 
are usually four well-marked spot belts, one on each side of 
the equator, due to the expiring disturbance, and two in high 
latitudes, due to the one just beginning. 

242. Cause of Sun-Spots. — Absolute knowledge is wanting 
here. Numerous theories have been proposed; many of them 
are now refuted, and of those that remain no one can be said 
to be fully established. On the whole, perhaps, at present the 

THE SVN 215 

most probable view is that they are tlie result of eruptions, — 

not, however, that they are craters through which the eruptions 

break out, as was at one time thought. It is more likely that 

when an eruption takes place a hollow or " sink " results in the The theory 

neighboring surface of the photosphere, in which hollow the of eruptions. 

cooler gases and vapors collect. 

Another theory, first proposed by Sir John Herschel and now 
favored by Lockyer and others, is that the spots are formed, 
not by any action from within, but by cool matter descending Theory that 
from above, and probably of meteoric origin ; but it is not ^^^^^ ^^® ^^^® 

to mctGors. 

easy to reconcile this with the peculiar distribution of the 
spots upon the sun's surface, though it falls in well with their 

Faye considers them to be cyclones in the solar atmosphere 
somewhat analogous to terrestrial storms. 

In 1894 E. Oppolzer of Vienna proposed the newest theory, which Meteoro- 
attributes them to bodies of gas and vapor which, ascending from the logit-al 
polar regions, drift towards the equator and descend in the spot zones, ^©oiies. 
becoming warmed and dried by the descent, — just as is the case with 
descending currents in the earth's atmosphere. If he is right, the spots 
are actually hotter than the underlying photosphere, but less luminous 
because, being purely gaseous, they radiate less powerfully. 

243. Terrestrial Influence of Sun-Spots. — One correlation The terres- 
between sun-spots and the earth is perfectly proved. When the *^"^^^ ^^^^" 

^ . . , . ence of 

spots are numerous magnetic disturbances ("magnetic storms") sun-spots. 
are most numerous and intense, and, as Ellis has also showed, 
the regular daily variations of terrestrial magnetism are also 
greatest ; in many instances also (but by no means always) 
notable disturbances upon the sun have been accompanied by 
violent magnetic storms and electric earth-currents, with brilliant Correlation 
exhibitions of the aurora borealis, as in 1859 and 1883. The ^^ terrestrial 


fact of the connection between terrestrial magnetism and solar disturbances 
disturbances is beyond doubt, though the nature and mechanism ^'^^^ ^"^" 


of this connection is as yet unknown, — we do not know whether 



Question of 
effect upon 
the meteor- 
ology of the 

the solar disturbance causes the terrestrial, or whether both dis- 
turbances are due to some external influence. 

The dotted lines in Figs. 88 and 89 represent the magnetic 
"storminess" at the indicated dates, and its correspondence 
with the sun-s]3ot curve makes it impossible to doubt their 

It has also been attempted to show that solar disturbances 
are accompanied by effects upon the earth's meteorology^ — upon 
its temperature, barometric pressure, storminess, and amount of 
rainfall. It can only be said that the matter is still under 
debate. While some particular investigations appear to show 
a correspondence for a time, others contradict them. If, as is 
not antecedently improbable, some real connection exists, other 
disturbances so mask and distort the sun-spot effects that the 
evidence is thus far inconclusive, and it may be many years 
before the question can be finally decided. At present it is not 
certain whether the earth is warmer or cooler, more rainy or 
less so, at the time of sun-spot maximum. 

It is certain that sun-spots cannot produce any sensible effects 
by their direct action in diminishing the heat and light of the 
sun, since they never cover as much as one-thousandth part 
of the solar surface. There seems to be, however, at present, 
according to Halm, a slight balance of statistical evidence in 
favor of the belief that on the whole the temperature of the 
earth is really slightly higher at or near a spot minimum than at 
a maximum. 


THE SUK {Continued) 

The Spectroscope, the vSolar Spectrum, and the Chemical Constitution of the Sun — 
The Doppler-Fizeau Principle — The Chromosphere and Prominences — The 
Corona — The Sun's Light — Measurement of the Intensity of the Sun's Heat 
— Theory of its Maintenance — The Age and Duration of the Sun — Summary 
as to the Constitution of the Sun 

About 1860 the spectroscope appeared in the field as a new 
and powerful instrument of astronomical research, resolving at 
a glance many problems which before had seemed to be abso- 
lutely inaccessible to investigation. It is not extravagant to 
say that its invention has done almost as much for the advance- 
ment of astronomy as that of the telescope. 

It enables us to study the light that comes from distant Astronomi- 
obiects, to read therein a record, more or less complete, of their ^^^ impor- 

J ' _ ' _ , tance of the 

chemical composition and physical conditions, to measure the spectro- 
speed with which they are moving towards or from us, and some- scope, 
times, as in the case of the solar prominences, to see and observe 
at any time objects otherwise visible only on' rare occasions. 

244. The Spectroscope. — The essential part of the instrument 
is either a prism or train of prisms, or else a "diffraction grat- The essential 
ing-," which is merely a piece of sflass or speculum-metal, ruled "^®"^^®^ ^f 

P J L & r thespectro- 

with many thousand straight equidistant lines, from ten thou- scope: 
sand to twenty thousand in each inch. Either the prism or the ^^® pnsm 
grating performs the office of " dispersing" the rays of different 
wave-length and color. 

If with such a "dispersion piece," as it may be called (either 
prism or grating), one looks at a distant point of light, — a star, 
for instance, — he will see, instead of a point, a long streak, red 
at one end and violet at the other. If the object observed is 




not a point, but a line of light parallel to the edge of the prism 
or to the lines of the grating, then instead of a mere colored 
streak without width one gets a spectrum, — a colored hand or 
ribbon of light, — which may show transverse markings that 
will give the observer most valuable information. 

It is usual to form this line of light by admitting the 
The slit and light through a narrow slit (seldom more than ^l-g- of an inch 
collimator, ^{^q^ placed at one end of a tube which carries at the other 

Prism Spectroscope 


The view- 



Direct-Vision Spectroscope 
Fig. 90. — Different Forms of Spectroscope 

end an achromatic object-glass having the slit in its principal 
focus {Physics, p. 369). The rays from the slit after having 
passed the object-glass form a parallel beam, just as if they had 
come from a very distant object. This tube with slit and lens 
constitutes the collimator, so named because it is precisely the 
same as the instrument used with the transit-instrument to 
adjust its line of collimation. 

Instead of looking at the spectrum with the naked eye it is 
better in most cases to magnify it by using a small view-telescope 

THE SUN^ 219 

(so called to distinguish it from the large telescope to which the 
spectroscope is often attached). 

The instrument, therefore, as usually constructed and shown 
in the diagram (Fig. 90), consists of three parts, — collimator, 
prism or grating, and view-telescope, — although in the " direct- 
vision " spectroscope, shown in the figure, the view-telescope is 

Fig. 91, from The Smi^ by permission of Appleton & Co., The tele- 
represents a large " telespectroscope" (as the combination of ^p^^*^^" 
telescope and spectroscope is called) arranged for photographic 

245. The Formation of the Spectrum. — If the slit S be illumi- 
fiated by strictly "homogeneous light" (z.e., all of one wave- 
length), a single image of it will be formed. If the light is How the 
yellow, a yellow image will appear at Y (Fig. 90). If at the j^^^*^^"" '" 
same time light of a different wave-length and color — red, for 
instance — be also admitted, a red image will be formed at i?, and 
the observer will then see a spectrum with two bright lines, the 
lines being really nothing more than images of the slit. If violet The spec- 
lip'ht also is admitted, a third (violet) imasfe will be formed at ^^^^^a series 

^ . . of images of 

F, and the spectrum will show three bright lines. the slit. 

If the light comes from a luminous solid^ like the lime cylin- 
der of a calcium light, or the filament of an incandescent lamp, 
or from an ordinary gas or candle flame (in which the light- 
giving particles are really bits of solid carbon)^ rays of all 
possible wave-lengths will be emitted and pass through the slit. The con- 
and as a consequence we shall have an infinite number of these ^^^^^^^^^ 

.... spectrum. 

slit-images packed close together, like a picket-fence in which 
the pickets touch each other ; we then get what is called a con- 
tinuous spectrum., ranging in color from red at one end to violet 
at the other, but showing no transverse lines or markings. 

If the light comes, however, from an electric discharge between 
two metallic balls, or in a so-called Geissler tube, or from a Spectrum of 
Bunsen-burner flame charged with the vapor of some volatile ^^^s^^ ^^^^^- 



Fig. 91. — Telespectroscope, fitted for Photography 
From The Sun, by permission of the publishers 



metal, the spectrum will consist of a series of bright lines or 
bands of different colors and usually numerous. 

246. The Solar Spectrum. — If we look at sunlight^ either 
direct or reflected (as from a piece of paper or from the moon), 
we get a spectrum, continuous in the main but crossed by The solar 
thousands of dark lines, or missing slit-images, known as the spectrum 

^ ^ and its dark 

"Fraunhofer lines," because Fraunhofer was the first to map Fraunhofer 
them (in 1814). To some of the more conspicuous lines he ^^^®^- 
assigned letters of the alphabet which are still retained as 
designations : thus, A is a strong line at the extreme red end of 








! I 

^^^^^^H^H^R* >; 



; ■'"" f "^ 









i dH 


Fig. 92. — H and K Region of Solar Spectrum 
From photograph by Jewell, Johns Hopkins University 

the spectrum ; C, one in the scarlet ; D, one in the yellow ; F, 
in the blue ; and H and K are a pair at its violet extremity. 
Fig. 92 is from a photograph of a small portion of the violet 
region of the solar spectrum including the great H and K lines. 
The central strip is made by light from the very edge of the 
sun ; the strips above and below, by light from the center of 
the disk ; there are some notable differences in the appearance 
of some of the lines in the two cases. 

On the scale of the lower band of this photograph the whole 
of the visible part of the solar spectrum would be about 20 feet 
long. Our present maps of the spectrum contain more than ten 



The ultra- 
violet and 
of the 

Position of a 

line depends 
upon the 
of the ray 
to which it 
is due. 

of lines in 
on their 
ment and 
istics. ' 


thousand lines, some strong and heavy, others so fine as to be 
hardly visible, but each as permanent a feature of the spectrum 
as rivers and oities on a geographical map. 

The visible portion of the spectrum is by no means the whole, 
— only a small part of it, indeed. Above H and K lies a long 
" ultra-violet " region consisting of rays whose wave-length is 
too short to affect our eyes, but crowded with dark lines and 
accessible to photography. At the other end, below A, there is 
an " infra-red " region some twenty times as long as the visible 
spectrum and consisting of rays, which, while they bring us 
a large part of all the heat we receive from the sun, have 
wave-lengths too long to produce vision. A small part of this 
infra-red spectrum can be photographed, but most of it is 
accessible only to such heat-measuring instruments as Lang- 
ley's "bolometer" (General Astronomy/, Arts. 348 and 344). 
This region also is full of interspaces of exactly the same 
nature as the dark lines in the visible spectrum. 

The position of each line in the spectrum depends entirely on 
the wave-length or luminous pitch of the ray which produces it, 
or rather (since the line is dark) has been suppressed, and is 
missing. The significance of the lines depends upon their 
arrangement and characteristics, just as the "sense" of a printed 
page lies in the letters and their grouping. As to the colors of 
the spectrum, the spectroscopist generally pays no more attention 
to them than the geographer to the colors on his map. 

The explanation of the Fraunhofer lines remained a mystery 
for nearly fifty years, until cleared up by the discoveries of 
Kirchhoff and Bunsen in 1859. 

247. Principles upon which Spectrum Analysis depends. — 
These (subs tan tiall}^), as announced by Kirchhoff in 1859, are 
the three following : 

(1) A Continuous Spectrum is given by luminous bodies, 
which are so dense that the molecules interfere with each other 
in such a way as to prevent their free luminous vibration, i.e., 





by bodies which are solid or liquid, or, if gaseous, are under Jdyh Origin of the 
pressure. Such bodies emit a iumble of all possible wave- c<^°^^^^^^"s 

-^ J L spectrum. 

lengths and colors. 

(2) The spectrum of a luminous gas under low ^^ressitre is Origin of the 
discontinuous, made up of bright lines or hands, and these lines 
are characteristic; i.e., the same substance under similar condi- 
tions always gives the same set of lines and generally does so 
even under conditions considerably different ; but it may (and 
many gases do) give 
two or more differ- 
ent spectra when the 
circumstances differ 
too widely. 

(3) A gas or vapor 
absorbs from a beam 
of white light which 
passes through it pre- 
cisely those rags of 
which, when the gas is 
luminous, its own spec- 
trum consists. The 
spectrum of the trans- 
mitted light then 
exhibits a reversed 

power pro- 
portional to 

Fig. 93. — Reversal of the Spectrum 

spectrum, which shows upon a continuous background dark 
lines replacing the bright ones that characterize the gas. 

This principle of reversal is illustrated by Fig. 93. In front 
of the slit of the spectroscope is placed a spirit-lamp or a 
Bunsen burner, with a little bead of carbonate of soda in the 
flame, and if we add a little salt of thallium, we shall then get 
a spectrum showing the two principal yellow lines of sodium 
and the green line of thallium, — all three bright. If now a 
lime-light or an electric arc be put in action behind the flame, 
we at once get the effect shown at the bottom of the figure, — 

Reversal of 
lines shown 



lines due to 
of rays by 
the atmos- 
plieres of 
the sun and 

tion of 
existing in 
the solar 


a brilliant continuous spectrum crossed by three hlacJc'^ lines 
which exactly replace the bright ones. Insert a screen behind 
the lamp flame and the lines immediately brighten again. 

The explanation of the Fraunhofer lines., therefore., is that they 
are mainly due to the absorbing action of the gases and vapors of 
the solar atmosp)here upon the light transmitted through them from 
the liquid or solid particles which compose the clouds of the solar 
photosphere. Some of the dark lines of the solar spectrum, 
known a,s telluric lines., are, howevei', due to the gases and 
vapors of the earth's atmosphere, — to water vapor and oxygen 

248. Chemical Constituents of the Sun. — Numerous lines of 
the solar spectrum have been identified as due to the presence 
in the sun's atmosphere of known terrestrial elements in the 
state of vapor. 

To effect the comparison necessary for this purpose the 
observer's apparatus must be so arranged that he can confront 
the spectrum of sunlight with that of the substance to be 
examined, which must be brought into the gaseous condition.^ so 
that it can emit its characteristic spectrum of bright lines. 

In the case of those substances which volatilize at a compara- 
tively low temperature, as, for instance, sodium, calcium, thal- 
lium, and the alkaline metals generally, the flame of a spirit-lamp 
or Bunsen burner answers the purpose. A little piece of the 
metal or of one of its easily volatilized salts is inserted in the 
flame, and the bright lines or bands of its spectrum appear at 
once in the spectroscope. 

If this flame is not hot enough, that of the oxyhydrogen 
blowpipe used for the calcium light may answer. 

1 Their apparent darkening, however, when the brilliant light from the lime 
is transmitted through the flame, is only relative, not real. Their brightness is 
actually a little increased ; but the brightness of the background is increased 
immensely, making it so much brighter than the three lines that, contrasted 
with it, they look black, as does an electric arc when interposed between the 
eye and the sun. 



This failing, recourse is had to electricity. Most of the 
metals vaporize at once in the electric arc between carbon 
electrodes, but we may have to employ the still higher tem- 
perature of an electric spark produced between 
electrodes of the metal by an "induction coil"; 
and in passing it is to be noted that the spectrum 
of the metal produced by the spark usually pre- 
sents notable differences from the arc spectrum. 

Finally, if we have a permanent gas, say hydro- 
gen, to deal with, it is sealed up, usually much 
rarefied, in a glass Geissler tube (Fig. 94), 5 or 
6 inches long, with metallic electrodes at each 
end, by means of which electrical discharges can 
be passed through the gas. 

249. Method of comparing Spectra. — In order 
to effect the comparison, half the slit is covered 
with a little reflector, or a so-called " comparison 
prism" which reflects into it the sunlight, while 
the other half of the slit receives directly the light 
from the luminous vapor. Upon looking into 
the spectroscope the observer will have the two 
spectra, of the sunlight and of the metal, side by 
side, and can at once see what bright lines of the 
metallic spectrum do or do not exactly coincide 
with the dark lines of the solar spectrum. If he 
finds that every one of the conspicuous bright 
lines matches a conspicuous dark line, he can be 
certain that the substance exists as vapor in the 
sun's atmosphere. 

In such comparisons photography may be most effectively 
used instead of the eye. The slit of the spectroscope is so 
arranged that either half of its length can be used indepen- 
dently. An impression of the solar spectrum is then obtained 
by a few seconds' exposure to sunlight admitted through one 

tion of 
by the elec- 
tric arc 
and spark. 

detected by 
the coinci- 
dence of 
bright lines 
in their 
spectra with 
lines in 
spectrum of 
the sun. 

Fig. 94 

Geissler Tube 

Use of 
in making 
the com- 



List of 
detected in 
this way. 

half of the slit, which is then closed, and the room darkened. 
Immediately afterwards light from an electric arc containing 
the vapor of metal to be tested is admitted through the other 
half for a sufficient time. The plate, when developed, will then 
show the two spectra side by side. Fig. 95 is a half-tone repro- 
duction, on a reduced scale, of a negative made by Professor 
Trowbridge in investigating the presence of iron in the sun. 
The lower half is part of the violet portion of the sun's spectrum 
(showing dark lines as bright), and the upper half that of an elec- 
tric arc charged with the vapor of iron. In the original every 
line of the iron spectrum coincides exactly with a correlative in 


5 ^ 


Fig. 95 

the solar spectrum, though in the engraving some of the coinci- 
dences fail to be obvious. There are, of course, on the other 
hand, certain lines in the solar spectrum which do not find any 
correlative in that of iron, being due to other elements. 

250. Elements known to exist in the Sun. — As the result of 
such comparisons, first made by Kirchhoff, but since repeated 
and greatly extended by late investigators, a large number of 
our chemical elements have been ascertained to exist in the 
solar atmosphere in the form of vapor. 

Professor Rowland in 1890 gave the following preliminary 
list of thirty-six whose presence may be regarded as certainly 
established, and it is probable that further research will add a 
number of others. The elements are arranged in the list accord- 
ing to the intensity of the dark lines by which they are repre- 
sented in the solar spectrum ; the appended figures denote the 



rank which each element would hold if the arrangement had 
been based on the number instead of the intensity of the lines. 

the case of iron 

the number exceeds 

, two thousand. 

* Calcium, ii. 

* Strontium, 23. 

Copper, 30. 

* Iron, 1. 

* Vanadium, 8. 

* Zinc, 29. 

* Hydrogen, 22. 

* Barium, 24. 

* Cadmium, 26. 

* Sodium, 20. 

* Carbon, 1. 

* Cerium, 10. 

* Nickel, 2. 

Scandium, 12. 

Glucinum, 33, 

* Magnesium, 19. 

* Yttrium, 15. 

Germanium, 32. 

* Cobalt, 6. 

Zirconium, 9. 

Rhodium, 27. 

Silicon, 21, 

Molybdenum, 17. 

Silver, 3i. 

Aluminium, 25, 

Lanthanum, 14. 

Tin, 34. 

* Titanium, 3. 

Niobium, I6. 

Lead, 35. 

* Chromium, 5. 

Palladium, I8. 

Erbium, 28. 

* Manganese, 4. 

Neodymium, 13. 

Potassium, 36. 

An asterisk denotes that the lines of the element indicated appear often or 
always as bright lines in the spectrum of the chromosphere (Sec. 257). 

Helium was added in 1895, — peculiar in that it manifests its Exceptional 
presence, not by dark Fraunhofer lines, but only by bright lines ^^^® "^ 
in the spectrum of the chromosphere. Certain observations of 
Runge on lines in the infra-red portion of the spectrum seem to 
indicate that oxygen should also be included. 

It will be noticed that all the bodies named in the list, carbon 
and hydrogen alone excepted, are metals^ and that many of the 
most important terrestrial elements fail to' appear; nitrogen, 
chlorine, bromine, iodine, sulphur, phosphorus, and boron are 
all missing, and oxygen gives only faint and as yet uncertain 
indications of its presence. 

251. Unsafety of Negative Conclusions. — We must be cautious. Negative 
however, in drawingf negative conclusions. It continually hap- conclusions 

^ ^ _ . . unwar- 

pens that when a mixture of gases or vapors is examined with ranted. 
the spectroscope, certain ones only can be recognized; as long 
as these are present the others keep in hiding. Thus the 
presence of argon in atmospheric air cannot be detected by the 






The revers- 
ing layer. 

Reversal of 
the Fraun- 
hofer lines 
at the 
instant of 
or end of 
The flash 

spectroscope until nearly all the oxygen and nitrogen have been 
removed ; and the other new gases of the atmosphere, krypton, 
neon, and xenon, are still more difficult to deal with. 

It is quite conceivable also that the spectra of the missing 
elements may be, under solar conditions, so different from their 
spectra as presented in our laboratories that we cannot recognize 
them ; for it is now unquestionable that many substances under 
different conditions give two or more widely different spectra, 
— nitrogen, for instance. 

Lockyer thinks it more probable that the missing substances are not 
truly "elementary," but are decomposed or "dissociated" by intense heat, 
and so cannot exist on the sun, but are replaced by their components. 
He maintains, in fact, that none of our so-called " elements " are really 
elementary, but that all are decomposable and are to some extent actually 
decomposed in the sun and stars ; and some of them by the electric spark 
in our own laboratories. Granting this, many interesting and remarkable 
spectroscopic facts find easy explanation. At the same time the hypothe- 
sis is encumbered with serious difficulties and has not yet been finally 
accepted by physicists and chemists. 

252. The Reversing Layer. — According to Kirchhoff's theory, 
the dark lines are formed by the transmission of light emitted 
by the minute solid or liquid particles of which the photospheric 
clouds are supposed to be formed, through somewhat cooler 
vapors containing the substances which we recognize in the 
solar spectrum. If this be so, the spectrum of the gaseous 
envelope, which by its absorption causes the dark lines, should 
by itself show a spectrum of corresponding bright lines. 

The opportunities are rare when it is possible to obtain the 
spectrum of this gas stratum separate from that of the photo- 
sphere ; but at the time of a total eclipse, at the moment when 
the sun's disk has just been obscured by the moon and the 
sun's atmosphere is still visible beyond the moon's limb, the 
observer ought to get this bright line spectrum, if his spectro- 
scope is carefully directed to the exact point of contact. 



The actual observation was first made during the Spanish 
eclipse of 1870. The lines of the solar spectrum, which up to 
the time of the final obscuration of the sun had remained dark 
as usual (with the exception of a few belonging to the spectrum 
of the chromosphere), were suddenly reversed, and the whole 
field of view was filled with brilliant colored lines, which flashed 
out quickly and then gradually faded away, disappearing in two 
or three seconds, — a most beautiful thing to see. 

The natural interpretation of this phenomenon is that the 
dark lines in the solar spectrum are, mainly at least, produced 

K H HS Hv 

Fig. 96. — The Flash Spectrum 


by a very thin stratum close down upon the photosphere, since 
the moon's motion in three seconds would cover a thickness 
of only about 800 miles. It was not possible, however, to be 
certain from such a mere glance that all the dark lines of the 
solar spectrum were reversed. 

Several partial confirmations of the observation have since been visually 
obtained at eclipses, though none so complete as desirable ; but the photo- 
graphs of the "flash spectrum," as it is now called, obtained during the Photo- 
recent eclipses of 1896, 1898, 1900, and 1901, made with various forms of graphs of 

the " prismatic camera " (a camera of long focus, with a prism, a train ^^^^ 

of prisms, or a " grating " outside the object-glass), have fully corroborated 

it. Fig. 96 is a reproduction of one of the exquisite photographs of the 

flash spectrum obtained by Sir Norman Lockyer in India during the eclipse 



of 1898. The lines above (to the left of H and K) are in the invisible 
portion of the spectrum and are most of them due to hydrogen. Until 
these permanent records of the phenomena were obtained there was room 
to doubt whether the bright lines seen might not belong mainly to the 
spectrum of the " chromosphere " (Sec. 257), instead of being reversed 
Fraunhofer lines. 

Sir Norman Lockyer has never admitted the existence of any such tJiin 
" reversing layer," maintaining that a large proportion of the dark lines 
are formed only in the regions of lower temperature, high up in the sun's 
atmosphere, and not close to the photosphere, i.e., different lines of a given 
substance originate at very different elevations in the solar atmosphere. 

253, Sun-Spot Spectrum. — The spectrum of a sun-spot differs 
from the general solar spectrum, not only in its diminished 

l\ ' 

20 Th n30 


a I a 

Fig, 96 a. — Portion of Sun-Spot Spectrum 
From photograph of 1893 

Peculiarities brightness, but in the great widening and intensification of cer- 
of the spec- ^^^^^ dark lines and the thinning^, and sometimes the reversal, 

trum of a _ ^ 

sun-spot. of others, especially those of hydrogen and H and K of calcium, 

"the great twin brothers," as Miss Gierke calls them, which 

are also conspicuous in the solar prominences, and, we may 

remark in passing, are also always reversed in the spectrum of 

the faculse, appearing as thin bright lines running through the 

center of the wide, black, hazy-edged bands. The majority of 

the Fraunhofer lines are, however, as a rule, quite unaltered; 

and in the case of those substances which show widened lines 

in the spot spectrum, only a few of their lines are thus affected. 

Some substances which are very inconspicuous in the ordinary 

Vanadium solar spcctrum bccomc obtrusive in the sun-spot spectrum, — 

m sun-spots, ycinadium, for instance. Fig. 96 a is from a photograph of the 



yellowish-green portion of a sun-spot spectrum and exhibits 
very well the leading characteristics. 

The general darkness of the spectrum of a sun-spot, in the 
green portion at least, appears to be due to the presence of 
myriads of thin dark lines so closely packed, with here and 
there an interval, as to be resolvable only in instruments of high 
power. This indicates that the darkening is due, in part at 
least, to the absorptio7i of light b^ transmission through vapors, 
rather tlian to a diminution of the emissive power of the surface 
from which the light comes. 

254. Displacement and Distortion of Lines. — Sometimes in 
the spectrum of an active sun-spot or of a prominence certain 
lines are displaced and broken, 
as shown in Fig. 97. These 
distortions can be explained as 
due to the swift motion towards 
or from the observer of the 
gaseous matter, which by its 
absorption produces the line 
observed. In the case illus- 
trated in the figure hydrogen 

was the substance, and its motion was away from the earth at 
the rate of nearly SOO miles a second. 

The general principle upon which the explanation of such 
phenomena depends was first enunciated by the German physicist 
Doppler in 1842, and has turned out to be one of extreme 
importance and wide application. It is this : When the distance 
between the observer and a body which is emitting regular vibra- 
tions is increasing, then the number of vibratiojis received in a 
second is decreased and their wave-length, real or virtual, is 
correspondingly increased; and vice versa if the distance is 

Thus, in the case of recession, the pitch of an engine whistle 
suddenly drops when a whistling engine passes us and recedes; 

due to 
by vapors. 

ment and 
of lines. 




Fig. 97. — The C Line in the Spectrum of 
a Sun-Spot, Sept. 22, 1870 




Effect of 
motion upon 
position of 
lines in the 
The Dop- 

and a light-ray (say the particular ray which produces the C line 
in the spectrum of hydrogen) has its wave-length increased and 
its refrangibility, which depends upon its wave-length, dimifi- 
ished, if the luminous object is receding, so that the C line and 
all the other hydrogen lines are shifted toward the red end of the 
spectrum. This effect of motion on the lines of the spectrum 
was first pointed out by Fizeau in 1848, so that in its astro- 
nomical application the principle is now usually referred to as 
the '' Doppler-Fizeau " principle. 

Fig. 98 illustrates the principle. The lower strip is a small piece of the 
yellow portion of the spectrum of a star (imaginary) which is rapidly 
approaching the earth, the two conspicuous dark lines being the D^ and D2 
lines of sodium. The upper strip is the corresponding part of the spectrum 

of a flame or electric spark con- 
taining sodium vapor and show- 
ing its lines bright. The two 
spectra are confronted by a 
" comparison prism " (Sec. 249), 
and it is obvious that the lines 
of the star spectrum are shifted 
towards the blue end by about 
one fourth of the distance between the D lines, i.e., by about 1.5 units of 
wave-length on the Rowland scale (the unit is one ten-millionth of a milli- 
meter). As the wave-length of D^ is 5896 units (nearly), it follows from 
the formula of the next article that the imaginary star must have been 
rushing towards us at the rate of nearly 48 miles a second, — pretty fast, 
but several real stars are swifter. 

FiCx. 98 

giving rela- 
tion between 
the radial 
velocity of a 
object and 
the shift of 
lines in its 

255. Formula of the Doppler-Fizeau Principle. — While the 
reasoning upon which the principle rests is simple, a general 
theoretical treatment for light-waves is difficult. 

For the demonstration of the formulae given below, the reader 
is referred to Frost's translation of Scheiner's Astronomical 
Spectroscopy, Part II, Chapter II. 

If V is the velocity of light (186330 miles a second), r the 
speed with which the observer is receding from the object, s 
the speed with which the source of light itself is receding., \ 

THE SUN 233 

the normal wave-length of the given line in the spectrum, and 
X' the apparent wave-length as affected by the two motions, 
we have the equation : 

X'=XX^-^. (1) 

V — r 

Subtracting X from both sides of the equation, we get 

V-X, or AX = X^^^, (2) 

V — r 

which holds for all velocities, great or small. 

Since, however, in all ordinary cases r is insignificant as com- 
pared with F, it may be dropped in the denominator, and we have 

AX = X X 


Finally, putting v for r + s, the total rate at which tlie distance 
between the object and the observer is iyicreasing^ we have 

AX V rr ^^ /QX 

— = —, or V = V X — , (3) 

XV X ^ ^ 

which is the usual formula employed in computing "motion in 
the line of sight " (or " radial velocity," as it is now usually 
called) from observations of the shift of lines in the spectrum. 
When the distance is decreasing, v becomes negative, and 
also AX, indicating a diminution of wave-length and a cor- 
responding shifting of the line towards the blue end of the 
spectrum. At present motions of less than half a mile per 
second can be detected by the spectroscopes which are used in 
studying stellar spectra. 

256. Other Causes of Displacement of Spectrum Lines. — It other causes 
has been recently (1895) discovered by Humphreys and Mohler ^i^^chpro- 

Q.U.C6 3. 

at Baltimore that the position of a line in the spectrum of a somewhat 
luminous vapor may also be shifted in a somewhat similar similar 


manner toivards the red by great increase oi pressure, — a shift of lines 
pressure of 180 pounds to the square inch producing as great in the 
a displacement as a receding rate of some 2 miles a second ; 
but the shift varies for different lines in the spectrum and does 
not follow the same law as in the case of motion. 




In 1900 Professor Julius of Utrecht demonstrated how an 
apparent shift of spectrum lines may also follow from what is 
called "anomalous refraction" in the sun's atmosphere near 
sun-spots and solar prominences ; and Michelson in a still more 
recent paper shows how rapid changes of density in the medium 
through which light comes to us may produce similar effects. 
It is quite possible, therefore, that some of the phenomena which 
have hitherto been explained on the Doppler-Fizeau principle 
as indicating tremendous velocities of moving matter may, on 
further examination, receive a different interpretation. 

257. The Chromosphere and Prominences. — Outside the photo- 
Thechromo- Sphere lics the chromosphere^ of which the lower atmosphere, or 
"reversing layer," is only the densest and hottest portion. This 
chromosphere, or " color sphere," is so called because it is bril- 
liantly scarlet, owing the color to hydrogen, which is its main, 
or at least its most conspicuous, constituent. The spectroscope 
shows it to be principally composed of hydrogen, helium, and 
calcium vapor. In structure it is like a sheet of flame over- 
lying the surface of the photosphere to a depth of from 5000 to 
10000 miles, and as seen through the telescope at a total eclipse 
of the sun has been aptly described as like " a prairie on fire." ^ 

At such a time, after the sun is fairly hidden by the moon, a 
number of scarlet star-like objects are usually seen blazing like 
rubies upon the contour of the moon's disk. In the telescope they 
look like fiery clouds of varying form and size, and, as we now 
know, they are projections from the chromosphere, or isolated 
clouds of chromospheric material. They were called prominences 
or protuberances, as a sort of non-committal name, while it was 
still uncertain whether they were appendages of the sun or of 
the moon. 

1 There is, however, no real burning in the -case, i.e., no chemical combina- 
tion going on betiveen the hydrogen and some other element like oxygen. Tlie 
hydrogen is too hot to burn in this sense, the temperature of the solar surface 
being above that of dissociation, — so high that any compound containing 
hydrogen would there be decomposed. 

The promi- 
nences or 



eons con- 
stitution de- 
by the spec- 
troscope in 

They were first proved to be solar during the eclipse of 1860, 
by means of photographs which showed that the moon's disk 
moved over them as it passed across the sun. Fig. 99 is from 
a photograph of the eclipse of April, 1893, by Schaeberle. 

Their real nature as clouds of incandescent gas was first 
revealed by the spectroscope in 1868, during the Indian eclipse Their gas 
of that year. On 
that occasion nu- 
merous observers 
recognized in 
their spectrum the 
bright lines of hy- 
drogen along with 
another conspicu- 
ous yellow line, 
at first wrongly at- 
tributed to sodium 
but afterwards to 
a hypothetical 
element then un- 
known in our lab- 
oratories and pro- 
visionally named 
"helium," its yel- 
low line being known as Dg (D^ and D2 being the sodium lines 
which lie close by). 

Helium was discovered as a terrestrial element in April, 1895, by Dr. The identifi 
Ramsay, one of the discoverers of argon. In examining the spectrum of cation of 
the gas extracted from a specimen of cleveite, a species of pitch-blende, he 
found the characteristic D3 line along with certain other unidentified lines 
which appear in the spectrum of the chromosphere and prominences. The 
same gas has since been found in a number of other minerals and mineral 
waters and also in meteoric iron. Its density turns out to be about double 
that of hydrogen, but less than that of any other known element, and it resists 
liquefaction more stubbornly than any other gas, — indeed, it is the only 

Fig. 99. — Prominences, 1893 

heUum as a 





with the 
without an 

one not yet subdued, excepting possibly some of the new gases (neon, etc.) 
not yet obtained in suiEcient quantity to permit investigation on this line. 
Chemically, it is extremely inert, refusing to enter into combination with 
other elements (as hydrogen does so freely), and therefore exists on the earth 
only in minute quantities. It seems, however, to be abundant in certain 
stars and nebulge, w'here its lines are conspicuous along with those of 
hydrogen. The Dg line is not the only helium line, but the chromosphere 
spectrum contains at least three others that are always observable, besides 
a dozen or more that occasionally make their appearance. 

The H and K lines of calcium are also, like those of hydrogen and 
helium, always present as bright lines in the chromosphere ; and several 
hundred lines of the spectra of iron, strontium, magnesium, sodium, etc., 

,r , XT have been observed in it now 

H and He 

and then. Fig. 100 shows the 
appearance of the calcium 
lines in the chromosphere spec- 
trum, and also the hydrogen 
line (He), which is close to 
the H line, as well as H^, to 
the left of K. 

258. The Prominences 
and Chromosphere observ- 
able at Any Time with 
the Spectroscope. — Dur- 
ing the eclipse of 1868 Janssen was so struck with tlie bright- 
ness of the hydrogen lines in the spectrum of the prominences 
that he believed it possible to observe them in full daylight, and 
the next day he found it to be so. He also found that by a 
proper management of his instrument he could make out the 
forms and structure of the prominences which he had seen 
the day before during the eclipse. Lockyer, in England, a few 
days later, but quite independently, made the same discovery 
and ascertained that the prominences were mere extensions 
from a hydrogen envelope completely surrounding the sun, and 
it was he Avho gave to this envelope the now familiar name of 
" chromosphere." His name is always, and justly, associated 

Fig. 100, — H and K Lines in ChromosiDhere 
^: Spectrum 



with that of Janssen as a co-discoverer. A little later Huggins 
showed that by simply opening the slit of the spectroscope the 
form and structure of the prominence, if not too large, could 
be observed as a whole, and not merely by piecemeal as before. 
Within the last few years it has become possible even to pho- 
tograph them by an instrument called a Spectroheliograph. 

259. How the Spectroscope enables us to see the Chromosphere 
and Prominences without an Eclipse. — The reason why we can- 
not see them by simply screening off the sun's disk is that 
the brilliant illumination of our own atmosphere near the sun 
drowns them out, as daylight does the stars. 

When we point the telespectroscope so that the sun's image 
falls as shown in Fig. 101, with its 
limb just tangent to the edge of the 
slit, then, if there be a prominence at 
that point, we shall get two overlying 
spectra: one, the spectrum of the illu- 
minated air ; the other, superposed upon 
this background, is that of the promi- 
nence itself. Now the latter is a spec- 
trum consisting of bright lines, or, if 
the slit be opened a little, of bright 

images of whatever part of the prominence may fall between 
the jaws of the slit, and the brightness of these lines or images 
is indep evident of the dispersive power of the sjjectroscope ; 
increase of dispersion merely sets the images farther apart, with- 
out making them fainter (except as light is lost by the trans- 
mission through a greater number of prisms). The spectrum of 
the aerial illumination, on the other hand, is that of sunlight, — 
a continuous spectrum showing the usual Fraunhofer lines ; and 
this spectrum is made faint by great dispersion. Moreover, it 
presents dark lines or spaces just at the very places in the 
spectrum where the bright images of the prominences fall. 
They therefore become easily visible. 

tion of the 
principle by 
which the 
makes the 
visible. It 
reduces the 
of the back- 
ground, but 
not that of 
the promi- 

Fig. 101. — Spectroscope Slit 
adjusted for Observation 
of the Prominences 


maj^ual of astroxomy 


A grating of ordinary power attached to a telescope of no more than 
2 or 3 inches aperture gives a very satisfactory view of these beautiful 
and interesting objects. The red image, which corresponds to the C line 
of hydrogen, is by far the best for visual observations. When the instru- 
ment is properly adjusted, the slit slightly opened, and the image of 
the sun's limb brought exactly to its edge, the observer at the eyepiece 
of the spectroscope will see things about asj^e have attempted to represent 
them in Fig. 102, — as if he were looking at the clouds in an evening sky 
from across the room through a slightly opened window blind. 

260. Different Kinds of Prominences. — The prominences may 

be broadly divided into two classes, — the " quiescent " or " dif- 
fuse," and the " eruptive " or, as Secchi calls them, the " metal- 
lic," because they show in 
their spectrum the lines of 
many of the metals in addi- 
tion to the lines of hydrogen, 
helium, and calcium. 

Tlie Quiescent Prominences 
are immense clouds, often 
from 50000 to 100000 miles 
in height and of correspond- 
ing horizontal dimensions, 
either resting directly upon 
the chromosphere as a base, 
or connected with it by stems 
and columns, as shown in 
Fig. 103 A^ though in some 
cases they appear to be entirely detached. They are not very 
brilliant and ordinarily show no lines in their spectrum except 
those of hydrogen and helium and H and K of calcium (which 
are often doubly reversed, as shown in Fig. 100) ; nor are 
their changes usually rapid, but they often continue sensibly 
unaltered sometimes for days together, i.e.^ as long as they 
remain in sight in passing around the limb of the sun. All 
their forms and behavior indicate that, like the clouds in our 

Fig. 102. — The Chromosphere and Promi 
nences seen in the Spectroscope 



own atmosphere, they exist, and float, not in a vacuum, but in 
a medium having a density comparable with their own, though 
not giving bright lines in its spectrum, and for that reason not 
visible in the spectroscope. They are found on all portions of 
the sun's disk, not being confined to the sun-spot zones. 

The Eruptive Promiyiences, on the other hand, appear only Eruptive, or 
in the spot zones, and as a rule in connection with active spots, "i^t^ii^^'' 

... promi- 

Their spectrum usually contains the bright lines of various nences 

metals, magnesium and iron being especially conspicuous, and 

sodium not infrequent. They origi- ^ 

nate not in the spots themselves, but 

in the disturbed faculous region just 

outside. Ordinarily they are not very 

large, though very brilliant; but at 

Prominences, Sept. 7, 1871, 12.30 p.m. 

Same at 1.15 p.m. 

Fig. 103. — A Solar Explosion 

times they become enormous, in one instance under the writer's 

own observation reaching an elevation of more than 400000 

miles. They are most fascinating objects to watch, on account 

of the rapidity of their changes. Sometimes the actual motion of Rapidity of 

their filaments can be perceived directly, like that of the minute- ^^^^"^g®- 

hand of a clock, and this implies a velocity of at least 250 miles 

a second in the moving mass. In such cases the lines of the 

spectrum are also, of course, greatly displaced and distorted. 

Fig. 103 represents a prominence of this sort at two times, separated by 
an interval of three quarters of an hour. The large quiescent prominence 




of pronii 

raphy of 
by simple 

of Fig. A was completely blown to pieces, as shown in B, by the "explo- 
sion," as it may be fairly called, which occurred beneath it, — the first case in 
which such a phenomenon was actually observed. See also Fig. 104, show- 
ing successive stages of an eruptive prominence photographed at Professor 
Hale's private observatory in 1895. A considerable number of similar 
occurrences have been recorded by various observers during the last thirty 
years, but they are by no means every-day affairs. 

The number of prominences of both kinds visible at one time on the 
circumference of the sun's disk ranges from one or two to twenty-five or 
thirty ; the eruptive prominences being numerous only near the times of 
sun-spot maximum. 

261. Photography of Prominences ; the Spectroheliograph. — 

It is possible, but not very satisfactory, to photograph a small 

10.34 10.40 10.58 . 

Fig. 104. — Photographs of Prominences, March 25, 1895 

After Hale 

raphy by 
the spectro- 

prominence by the same arrangement as for visual observation, 
merely putting a sensitive plate at the focus of the "view- 
telescope " (Sec. 244), using the F line of the spectrum, or, 
better, the K line, instead of the red C, which requires too 
long an exposure. 

A much better plan is to use a " spectroheliograph," — inde- 
pendently devised by Professor Hale of Chicago and Deslandres 
of Paris. A detailed description would take too much space, 
but the essential feature is a narrow slit moving in front of the 
sensitive plate in exact accordance with the motion of the 



collimator slit ; so that as the latter is moved across the image of 
the prominence, while the latter moves before the plate, the 
bright K line of each portion of the prominence shines through 
upon the plate and so photographs the object in sectio7is, by its 
"K-light," if the ex- 
pression may be 

Fig. 104 is from a nega- 
tive thus made at Pro- 
fessor Hale's private 
Kenwood observatory on 
March 25, 1895, with 
the then newly invented 
spectroheliograph, — three 
exposures on an ascend- 
ing prominence at inter- 
vals of eight and eighteen 
minutes. During this time 
the height of the promi- 
nence increased from 
135000 miles to 281000. 
Fig. 104 A exhibits the 
whole sun with its spots 

and faculae and the surrounding chromosphere, — made by the same in- 
strument. A screen covers the sun's image while the chromosphere and 
prominences are first photographed by a slow motion of the slit, and then, 
the screen being removed, the slit is drawn back rapidly across the sun's 
image, thus producing the picture of the solar surface. 

262. The Corona. — The corona is a halo or glory of light which The corona; 
surrounds the sun at the time of a total eclipse and has been ^^^ general 


known from remote antiquity as one of the most beautiful and 
impressive of all natural phenomena. The portion near the 
sun is dazzlingly bright and of a pearly tinge, which contrasts 
finely with the scarlet prominences. It is made up of filaments 
and rays which, roughly speaking, diverge radially, but are 
strangely curved and intertwined. At a little distance from 

Fig. 104 a. 

Spectroheliogram of Entire Sun 
After Hale 



between the 
corona at a 
and at a 

the edge of the sun the light becomes more diffuse, and the 
outer boundary of the corona is not very well defined, though 
certain dark rifts extend through it clear to the sun's surface. 

Often the filaments are longest in the sun-spot zones, giving 
the corona a roughly quadrangular form. This seems to be 

Fig. 105. — Corona of 1871 

specially the case in eclipses which occur near the time of a 
sun-spot maximum. In eclipses which occur near the sun-spot 
minimum^ on the other hand, the equatorial rays predominate, 
forming streamers, sometimes fan-like and sometimes pointed, 
extending several millions of miles from the solar surface. 
Near the poles of the sun there are usually tufts of sharply 
defined threads of light, which curve both ways from the pole. 



The corona varies greatly in brightness at different eclipses, Light given 
accordinp- to the apparent diameter of the moon at the time, ^^ ^ 

n irir ^ corona. 

and with the sun-spot period. Its total light is always at least 
two or three times as great as that of the full moon. 

Drawings and Photographs of the Corona. — There is very great diffi- Drawings 
culty in getting accurate representations of this phenomenon. The two ^^^^ photo- 
or three minutes during which only it is usually visible at any given ^^ 
eclipse do not allow time for trustworthy hand-work ; at any rate, draw- 
ings of the same corona made even by good artists, sitting side by side, 











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V 'i^BH 



^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^ ^^^E -^^^^^1 

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^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^H^^^''~ ^^^^^^^^^ 


^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^iu i^ii^ir r t' &_%^a 


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^^^^^^^^^^^K.5? >->^P^I^^^^^^H 






Fig. 106. — Corona of Eclipse of 1900, Wadesboro, N.C. 

differ very much, — sometimes ridiculously. Photographs are better, so 
far as they go, but hitherto they have not succeeded in bringing out 
many details of the phenomenon which are easily visible to the eye ; nor 
do the pictures which show well the outer portions of the corona generally 
bring out the details near the sun's limb, though an ingenious device of 
Burckhalter, which, by a whirling screen of peculiar form in front of the 
sensitive plate, gives a much longer exposure to the outer regions than to 
the parts near the sun, has greatly improved the results. 

Fig. 105 is copied from an engraving made by combining several photo- 
graphs of the eclipse of 1871. Fig. 106 is reduced from a drawing 
by Wesley of the corona of the eclipse of May, 1900, made from a 



combination of the sketches and photographs obtained by the members of 
the various eclipse parties of the British Astronomical Association. It is 
an admirable representation of what the writer saw at Wadesboro, N.C., 
except that the curved wings on the west and the long, pointed, eastern 
streamer could all be traced much farther by the eye, — to a distance fully 
three and a half times the moon's diameter, or at least 3 000000 miles. 

The spec- 
trum of the 
corona. The 
green line, 
X = 5304. 

probably a 
gas of ex- 
tremely low 

263. Spectrum of the Corona. — The characteristic feature of 
the visual spectrum is a bright green line which lies very near, 
and was long supposed to be the " reversal " of, a dark line in 
the solar spectrum, — known as the " 1474 line," because its 
position is at 1474 on the scale of Kirchhoff's map of the spec- 
trum, generally used in 1869, when the corona line was first 
discovered. This identification, for which the writer was 
mainly responsible, turns out, however, to be erroneous, the 
wave-length of '' 1474 " being about 5317 on Rowland's scale, 
while the wave-length of the real corona line, as first discovered 
from the photographs in 1898 (and since confirmed), is 5304. 

The " 1474 " line (probably of iron) is always present in 
the chromosphere spectrum as a bright line, and at an eclipse 
when the corona first appears it is much the brightest line in 
the green part of the spectrum ; but, as we now know, it fades 
out shortly, while the true corona line, which is much fainter, 
remains, of course, visible through the whole totality. 

The unknown substance which produces this corona line has 
been provisionally named "coronium," just as ''helium" was 
named twenty-seven years before it was identified as a terres- 
trial element. It is probably a gas of extremely low density, — 
perhaps even lighter than hydrogen. 

Besides this conspicuous green line there are several others, 
— five at least and probably more, — all in the violet or ultra- 
violet, all probably due to the same substance, and showing, 
like the principal line, but much more faintly, as rings on 
photographs made by the prismatic camera during the recent 


The hydrogen and helium lines, and H and K of calcium, Hydrogen 
have also been photographed as bright lines during eclipses and ^^^^Jieiiuin 
attributed to the corona ; the later observations prove, however, the corona, 
that they are not really coronal, but are due to reflection (in our 
own atmosphere) of light coming from the chromosphere and 

The light of the corona is distinctly polarized : on one or Part of the 
two occasions observers have also reported in its visual spec- ^^gh^ofthe 

■^ _ ^ corona due 

trum the presence of dark Fraunhofer lines, and these have now to reflection, 
at last been successfully photographed by the Lick observers 
during the Sumatra eclipse of 1901. The corona therefore 
unquestionably contains some matter which reflects sunlight. 

264. Nature of the Corona. — The corona is proved to be a The corona 
true solar appendag-e and not a mere optical phenomenon, nor ^ ^°^^\ 

■^ -^ *=* ^ ^ appendage. 

due to either the atmosphere of the earth or moon, by two 
unquestionable facts : first, its spectrum as described above is 
not that of mere reflected sunlight, but of a glowing gas ; 
and second, photographs of the corona made at widely different 
stations on the track of an eclipse show, in the main, details 
that are identical as seen at stations thousands of miles apart, 
and exhibit the motion of the moon across the coronal filaments. 
In a sense, then, the corona is a phenomenon of the sun's 
atmosphere, though this solar " atmosphere '.' is very far from Not related 
bearing^ to the sun the same relations that are borne towards the ^^ ^^® ^^° 

*=* ^ ^ as our own 

earth by the air. The corona is not a simple spherical envelope atmosphere 

of gas comparatively at rest and held in equilibrium by gravity, ^^ related to 
but other forces than gravity are dominant in it, and matter 
that is not gaseous probably abounds. 

Its phenomena are not yet satisfactorily explained and remind Specuia- 

us far more of auroral streamers and comets' tails than of any- ^^^^^ ^^ *° 

_*^ its nature 

thing that occurs in the lower regions of our terrestrial and the 
atmosphere. Indeed, there are many features which warrant forces which 

.... , determine 

something more than a mere suspicion that it is more or less its form, 
analogous to Roentgen and cathode rays, due, as Professor 



Its extreme 

No method 
yet found 
for observ- 
ing the 
corona at 
other times 

Bigelow suggests, to io7is driven off from the molecules of the 
solar gases and controlled in their motions by electric and 
magnetic forces emanating from the sun. (See also Sec. 502.) 
That the corona is composed of matter excessively rarefied 
is shown by the fact that in a number of cases comets are 
known to have passed directly through it (as, for instance, in 
1882) without the slightest perceptible disturbance or retarda- 
tion of their motion. Its mean density must, therefore, be 
almost inconceivably less than that of the best vacuum we are 
able to make by artificial means. 

Numerous attempts have been made to find a way of observing the 
corona without an ecHpse, but thus far without any certain result. The 
spectroscopic method, which succeeds with the chromosphere and promi- 
nences, fails with the corona, because the lines of its spectrum are not 
bright enough ; and, moreover, there are, in the ordinary solar spectrum, no 
dark lines to match them and help in forming a background. Further- 
more, since the streamers of the corona are probably not entirely gaseous, 
but partly of dust-like constitution (giving, therefore, the spectrum of 
reflected sunlight), they at least would not be observable by this method. 

The sun's 


265. The Sun's Light. — By photometric methods {Physics, 
p. 328) we can compare the sun's light with that of a " standard 
candle " [Physics, p. 327), and we find that the sun gives us 
1575 billions of billions (1575 x 10^^) times as much light as a 
standard candle would give at that distance. 

Experiment shows that when the sun is overhead sunlight 
illuminates a white surface about 65000 times as brightly as a 
standard candle at one meter distance, or certainly not less than 
70000 times, if we allow for the absorption of sunlight in our 
air. Multiplying this 70000 by the square of the sun's distance 
in meters (15 x 10^^)^, we get the sun's " candle-power " as 
stated above. But the determination cannot claim any minute 
accuracy, because of the continual variations in the puritj^ of 

THE SUN 247 

our atmosphere and the difficulty in determining the losses of 
the sunlight before it reaches the photometric screen. 

The light received from the sun is about 600000 times that Sunlight 
of the moon, about 7000 000000 times that of Sirius, the "^.';^Pf"f^ 

^ ^ with light 

brightest of the fixed stars, and about 50000 000000 times that from other 
of Vega or Arcturus. * bodies. 

The intensity of sunlight is the amount of light emitted by 
each square unit of its surface, — a very different matter from 
its total quantity. From the best data at present obtainable 
(only rather rough approximations are possible) the sun's 
surface appears to be about 190000 times as bright as a intensity of 
candle flame, and about 150 times as bright as the calcium sunlight, 
light. The brightest part of the electric arc-light — its " crater" 
— comes nearer to the brilliance of the solar surface than 
anything else we know of, being from one half to one fourth as 

266. Relative Brightness of Different Parts of the Sun's 
Disk. — As already stated (Sec. 233), the sun's disk is brightest 
at the center, but the variation is very slight until near the Darkening 
edo^e, where the brightness falls off very rapidly, so that at the °^ ^^^ ^"^^^ 

, \ . . ° . . of the sun 

limb itself it is not more than one third of the brightness at and modi- 
the center. The color is modified also, vergingf towards choco- fixation 

of color. 

late, because the blue and violet rays are much more affected 
than the red and yellow ; this is the reason why the darkening 
at the limb of the sun is so much more conspicuous in the 
photographs than in the telescope. 

The darkening is unquestionably due to the general absorp- 
tion of light by the lower parts of the sun's atmosphere, though The sun 
it is difficult to determine iust how much the sun's brightness stripped of 
is diminished by it. Different estimates vary considerably, but p^ere 
according to Langley, if the sun's atmosphere were removed, we would be 
should receive from two to five times as much light as now, and, j^^^^i^ ^^^^ 
moreover, its tint would be strongly hlue^ more blue even than brilliant, 
that of an electric arc. 



The sun's 
heat: its 
in calories. 
The solar 

The engi- 
and the 
small calory, 
or calorette. 

obtained for 
the solar 
range from 
19 to 40. 

Method of 
the solar 

267. The Quantity of Solar Heat; the Solar Constant. — TAe 

solar constant is the number of heat-^nits which a square unit of 
the eartKs surface^ unprotected hy any atmosphere^ and exposed 
perpendicularly to the suns rays, would receive in a unit of time. 
(It is a somewhat doubtful assumption that this quantity is 
really "constant" under all the changing conditions of the 
solar system ; but if not, the fact has not yet been shown by 
observation.) The heat-unit most used at present, by engineers 
at least, is the calory, which is the amount of heat required to 
raise the temperature of one kilogram of water 1° C. A smaller 
unit, only j-^-q-q part as great, is much used in scientific work, 
substituting in the definition a gram of water for the kilogram. 
This heat-unit is called the "small calory," — it might perhaps 
be named the "calorette." 

As the result of the best observations thus far made, it is 
found that the solar constant is between twenty-five and thirty 
engineer s calories per square meter per minute, or 3.0 "calo- 
rettes" per square centimeter per minute.^ 

In what follows we shall use 30 as the solar constant, 
although Scheiner in 1899 sets it still higher, — at 40. The 
different determinations, since that first made by Pouillet and 
Herschel in 1838, range all the way from 19 to 40, — an indica- 
tion of the extreme difficulty of the subject. At the earth's 
surface a square meter seldom actually receives more than 
fifteen calories in a minute, fully fifty per cent being lost, or 
diverted, in its passage through the atmosphere. 

268. Method of determining the Solar Constant. — The princi- 
ple is simple, though the practical difficulties are very great, 
and so far have prevented any high degree of accuracy. 

The determination is made by admitting a beam of sunlight, of 

1 There would be some advantage in stating the solar constant on the C.G.S. 
system, by dividing 3.0 by 60, the number of seconds in a minute. According 
to this, the solar constant equals 0.050 C.G.S.., or ^l of a "calorette," received 
on a square centimeter in one second. 

and uncer- 

THE SUN 249 

hnown cross-section^ to fall upon a known quantity of water ^ for 
a known time, and measuring the consequent rise of temperature. 

It is necessary, however, to determine and allow for all heat 
lost hy the apparatus during the experiment or received from 
other sources, and especially to take into account the effect of 
atmospheric absorption. This is the most difficult and uncer- Difficulties 
tain part of the operation, since the absorption is continually 
changing, — capriciously with the meteorological conditions, and 
regularly with the changing altitude of the sun. 

It should be noted that the rays absorbed by the atmosphere, 
though diverted from the instrument which is endeavoring to 
measure their amount, are 7iot lost to the earth. The air is 
illuminated and warmed by them, and the earth gets the bene- 
fit of the effect at second hand, so to speak. 

For a description of the pyrheliometer and actinometer, with which the 
heat radiation is measured, and of the bolometer, with which the percent- 
age of loss is determined for rays of differing wave-length, the student is 
referred to the General Astronomy, Arts. 340-345. 

269. The Solar Heat at the Earth's Surface expressed in Terms Soiar heat 
of Melting Ice. — Since it requires 79i calories of heat to melt expressed in 

_ ^ terms of 

a kilogram of ice with a specific gravity of 0.92, it follows that, melting ice. 
taking the solar constant at 30, the heat received from a verti- 
cal sun would melt in an hour a sheet of ice 24.7 millimeters, 
or very nearly an inch, in thickness. From this it is easily 
computed that the amount of heat received by the earth from 
the sun in a year is sufficient to melt a shell of ice 225 feet 
thick on the earth's equator, or 177 feet over the earth's entire 
surface, if the heat were equally distributed over all latitudes. 

270. Solar Heat expressed as Energy. — Since according to Expressed 
the known value of the "mechanical equivalent of heat" as energy 
{Physics, p. 175) a horse-power (33000 foot-pounds per minute) i^orse- 
can easily be shown to be equivalent to about 10.7 calories per power, 
minute, it follows that each square meter of the earth's surface 



The solar 

Solar radi- 
ation at 
the sun's 

perpendicular to the sun's rays ought to receive about 2.8 horse- 
power continuously. Atmospheric absorption cuts this down to 
about If horse-power, of which about one eighth can be realized 
by a suitable machine, such as Ericsson's solar engine (Fig. 107), 
which for several years was exhibited annually in New York. 

The solar heat was concentrated by the large reflector, 18 feet in diame- 
ter, upon the ioiler, which was a 6-inch iron tube. A "head" at the upper 
end (removed when the photograph was taken) received the steam, and a 

pipe connected it with 
the i-horse-power en- 
gine shown under- 
neath. When the sun 
shone it worked well. 

The energy an- 
nually received 
from the sun by the 
whole of the earth's 
surface aggregates 
nearly 100 mile- 
tons to each square 
foot. That is, the 
average amount of 
heat annually re- 
ceived by each 
square foot of the 
earth's surface, if utilized in a theoretically perfect heat engine, 
would hoist nearly 100 tons to the height of a mile. 

271. Solar Radiation at the Sun's Surface. — If now we esti- 
mate the amount of radiation at the sun's surface itself, we 
come to results which are simply amazing. We must multiply 
the solar constant observed at the earth by the square of the 
ratio between 93 000000 miles (the earth's distance from the 
sun) and 433250 (the radius of the sun). This square is about 
46000. In other words, the amount of heat emitted in a minute 

Fig. 107. — Ericsson's Solar Enarine 

THE SUN 251 

by a square meter of the sun's surface is about 46000 times as 

great as that received by a square meter at the earth. Carrying 

out the figures, we find that this heat radiation at the sun's 

surface amounts to 1 4-00000 calories per square meter per 

minute; that if the sun were frozen over completely^ to a depth of 

64 feet, the heat emitted would melt the shell in one minute; Expressed 

that if a bridge of ice could be formed from the earth to the ^^ f ® 

^ melting. 

sun by a column of ice 2-^ miles square and 93 000000 long, 
and if in some way the entire solar radiation could be concen- 
trated upon it, it would be melted in one second, and in seven 
more would be dissipated in vapor. 

Expressing it as energy, we find that the solar radiation is Expressed 

nearly 130000 horse-power continuously for each square meter of ^^ ^^'^^" 
d 1 t/ ./ 2 power. 

the sun's surface. 

So far as we can see, only a minute fraction of the whole 
radiation ever reaches a resting place. The earth intercepts 
about 22 0^00 000 ^ ^^^^ ^^® other planets of the solar system Question as 
receive in all perhaps from ten to twenty times as much. Some- ^^ ^^^* 

1 , . . . . . becomes of 

thing like iqq gooooo seems to be utilized within the limits of heat radi- 
the solar system. As for the rest, science cannot yet give any ^^^d into 


certain account oi it. 

All the conclusions stated in the two preceding sections are 
based on the assumption that the sun radiates heat in all directions 
alike, whether the rays do or do not meet a material surface. 
No reason can be assigned why this assumption should not be 
true, but it cannot be said to be proved as yet, either by experi- 
ment or by the nature of the case. If it should ever be shown 
to be incorrect, certain difficulties in the theory of planetary 
evolution would be greatly mitigated. 

272. The Sun's Temperature; Effective Temperature. — As The temper- 
to the temperature of the sun's surface we have no exact knowl- ^*^^^® ^\ ^}^f 

■^ _ _ sun : widely 

edge, except that it must be higher than any artificial heat different at 
which we are able to produce. Indeed, it is only " by courtesy," different 

, 111 1 .11 points. 

so to speak, that the sun can be said to have a temperature. 



12000° F. 

The burning 

since the temperature at different elevations above and beneath 
the surface must differ enormously ; nor, probably, is it the 
same in a sun-spot as in the faculye, though we note in passing 
that observations indicate no systematic difference depending 
on position upon the sun's surface, i.e., on solar latitude and 

When we speak of the temperature of the sun we mean 
what is called the "effective temperature," i.e., the temperature 
that a surface of standard radiating power (lampblack is the 
standard) would require in order to radiate heat at the same 
rate as the sun. If the actual surface of the sun has a radiat- 
ing power inferior to that of the standard, as is probably the 
case, then the actual mean temperature must be higher than 
the effective, and vice versa. 

If we knew absolutely the law which connects the radiation 
rate of a surface with its temperature, we could compute the 
effective temperature from the solar constant. 

If we accept as correct Stefan's Law, which is borne out by 
the most trustworthy recent laboratory work, viz., that the rate 
of radiation is proportional to the fourth power of the absolute 
temperature,^ the sun's "effective temperature" comes out about 
7000° C. or 12000° F. 

The highest temperatures artificially obtained in the electric 
arc are in the neighborhood of 4000° C. 

The assumption of various other laws of radiation has led to 
a ridiculously wide range of computed solar temperatures, all 
the way from 1500° C, by Pouillet, to the millions of Secchi 
and Ericsson. And the result, as stated above, is doubtful by 
at least 500° C. 

273. The Burning Lens. — A most impressive demonstration 
of the intensity of the sun's heat lies in the fact that in the 
focus of a powerful burning lens all known substances melt and 

1 The absolute temperature is the temperature reckoned from the absolute 
zero, - 273° C. or - 449.4 F. 


THE SUN 253 

vaporize. Now at the focus of a lens the terr},perature can never 
more than equal that of the source from ivhich the heat comes. 
Theoretically, the limit is that temperature which would be 
produced by the sun's direct radiation at a distance such that 
the sun's apparent diameter would just equal that of the lens 
viewed from its focus. 

The temperature produced at F (Fig. 108) would, if there Highest 
were no losses, be iust the same as that of a body placed so temperature 

'' ^ ^ "^ ^ reached by 

near the sun that the sun's angular diameter equals LFL' . burning 
Now, in the case of the most powerful lenses hitherto made, 
about 4 feet in diameter, a body at the focus was thus virtu- 
ally carried to within about 240000 miles from the sun's surface 
(the distance of the moon from the earth), and here, as has been 
said, the most refractory 
substances succumb 
immediately. ^^-^ > 

A corroboration of 

the evidence of the 

burning lens is found ^m 108 

in the great extension 

of the solar spectrum into the ultra-violet region and in the 

penetrating power of the solar rays. Rays coming from a source 

of comparatively low temperature — from a stOve, for instance — 

are almost wholly absorbed by a plate of glass, while those of 

the sun pass almost without loss. 

274. Constancy of the Sun*s Heat. — It is an interesting ques- Constancy 
tion, as yet unanswered, whether the total amount of the sun's °^ *^® ^^^^'*^ 

heat: no 

radiation does or does not perceptibly vary. There may be sensible 

considerable fluctuations in the quantity of heat hourly received change for 
„, 1.11 1 1 1-1 ^^^ P^s* ^^^ 

irom the sun without our being able to detect them surely with thousand 

our present means of observation, but as far as ob^rvations go years, 

there is no evidence that the total amount varies very much. tempm-ary 

As to any steady, progressive increase or decrease of solar fluctuations 

heat, it is quite certain that no important change of that kind ^^^ ^^'^^^ .^" 



nance of the 
sun's heat. 

Not to be 
for by com- 
bustion or 
fall of 

due to slow 
of the sun's 

too small to 
be detected 
by observa- 
tion as yet. 

has been going on for the past two thousand years, because the 
distribution of plants and animals on the earth's surface is practi- 
cally the same as in the days of Pliny; it is, however, rather 
probable than otherwise that the general climatic changes which 
geology indicates as having formerly taken place on the earth 
— the glacial and carboniferous epochs, for instance — may ulti- 
mately be traced to changes in the sun's condition. 

275. Maintenance of Solar Heat. — One of the most interesting 
and important problems of modern science relates to the expla- 
nation of the method by which the sun's heat is maintained. 
We cannot here discuss the subject fully, but must content 
ourselves with saying, — 

(1) Negatively^ that the phenomenon cannot be accounted for 
on the supposition that the sun is a hot, solid, or liquid body 
simjjli/ cooling^ nor by combustion^ nor (adequately) bg the fall of 
meteoric matter on the su)i's surface, though this cause undoubt- 
edly operates to a limited extent. 

(2) Positively, the solar radiation can be accounted for on the 
hypothesis, proposed first by Helmholtz, that the sun is shrinking 
slowly but contiiiuously . It is a matter of demonstration that 
an annual shrinkage of about 300 feet in the sun's diameter 
would liberate heat sufficient to keep up its present observed 
radiation without any fall in its temperature. If the shrinkage 
were more than 300 feet, the sun would be hotter at the end 
of a year than it was at the beginning. 

It is not possible to exhibit this hypothetical shrinkage as a fact of 
observation, since this diminution of the sun's diameter would amount to 
a mile only in 17.6 years, and nearly eight thousand years would be spent 
in reducing it by a single second of arc. No change much smaller than 
V could be certainly detected even by our most modern instruments. 

We can only say that while no other theory yet proposed meets 
the conditions of the problem, this appears to do so perfectly, and 
therefore has high probability in its favor, especially as it appears to be 
a mere continuation of the process by which the present solar system was 

THE SUN 255 

276. Lane's Law. — It was first pointed out by Lane of Lane's Law: 
Washington in 1870 that a gaseous body, losing heat by radia- ^"^^^^^f 

^doj coil" 

tion and contracting under its own gravity, must rise in tempera- trading 
ture until it ceases to be a "perfect gas," either by beginning to ""^^^ ^^^ 
liquefy, or by reaching a density at which the laws of "perfect from loss of 
gases" cease to hold. In a mass of perfect gas the "work" heat, risesin 
due to its shrinkage (like the work done by a descending clock ^^^^j j^. 
weight) is more than sufficient to replace the loss of tempera- ceases to be 
ture due to its radiation, and it therefore becomes hotter. This ^„^^^ ^^ 


is not the case with a mass of solid or liquid, which, as it loses 
heat and begins to liquefy or solidify, diminishes in temperature 
as well as dimensions, and grows colder. 

It appears that in the sun at present the relative proportion 
of true gases and liquids (the droplets which form the pho- 
tospheric clouds) is such as to keep the temperature nearly 
stationary, — a condition which may endure for thousands of 


The now generally received opinion on this subject may be 
summed up substantially as follows : 

277. The Central Nucleus. — As to the condition of this we The central 
cannot claim certain knowledge, but many considerations lead 
to the conclusion that it is purely gaseous and has a temper- 
ature immensely higher than that of the solar surface even. 
But this central mass, while gaseous in that it follows essen- 
tially the characteristic laws of Dalton, Boyle, and Charles 
{Physics, p. 274), must be greatly condensed by the enormous 
pressure due to solar gravity; denser than water, and viscous, so 
to speak, like tar or pitch in resisting rapid motion within it. 

Certain phenomena, however, such as the tendency of photo- 
spheric disturbances (sun-spots and faculse) to break out repeat- 
edly in the same region (Sec. 237), suggest something like a 
quasi-solidity , sufficient to lead to the definite localization of 




The photo- 
sphere a 
cloud sheet. 

certain conditions at certain points below the photosphere ; and 
there are other phenomena which rather tend in the same direc- 
tion. Indeed, there are some who still hold to a solid or liquid 
nucleus for the sun. 

278. The Photosphere. — The photosphere is believed to be a 
sheet of luminous clouds^ constituted mechanically like terres- 
trial clouds, i.e.^ of minute solid or liquid particles floating 
in gas. These photospheric clouds are supposed to be formed 
(just as clouds of rain and snow are formed in our own atmos- 
phere) by the cooling and condensation of vapors at the solar 
surface where they are exposed to the cold of outer space, and 
they float in the permanent gases of the solar atmosphere in the 
same way that our own clouds do on our own atmosphere. 

We do not know precisely what materials constitute the 
photosphere, but naturally suppose them to be those indicated 
by the Fraunhofer lines, i.e.^ chiefly the metals, with earhon and 
its chemical congeners. 

of the cloud- 
sheet theory. 

theory of 
the photo- 

But this cloud theory of the photosphere is not without its difficulties. 
It is embarrassed by the fact that we know of no substance that remains 
solid or liquid, even at a temperature anywhere near that which seems to 
prevail at the solar surface. Carbon, perhaps the most refractory of known 
substances, vaporizes completely at a temperature of about 7000° F. Still 
the temperature of 12000° F. ascribed to the solar surface is only the 
"effective" average temperature, and possibly is not inconsistent with the 
hypothesis that the "granules" of the photosphere are due to local cool- 
ings caused by explosive expansion of vapors forced up from below by 
tremendous pressure into or through a gaseous envelope of much higher 

Some are disposed to evade the difficulty by invoking an electrical 
action of some kind, but as yet in too vague a manner to permit intelligent 

We merely mention an ingenious theory proposed a few years ago 
by Prof. A. Schmidt of Stuttgart, viz., that the photosphere is a purely 
optical phenomenon, a sort of mirage so to speak, the sun itself being 
entirely gaseous. The theory, based wholly on optical principles, has 
some strong points ; bat it ignores many spectroscopic facts, and the 

THE SUI^ 257 

fundamental laws of physics seem to make it certain that a globe contain- 
ing iron and the other metals in the state of vapor must inevitably form a 
photospheric shell of "cloud" in the outer portions exposed to radiation, 
thus "clothing itself with light as with a garment." 

279. The Solar Atmosphere ; the Reversing Layer, Chromo- The revers- 
sphere, and Prominences. — As has iust been said, the photo- ^J^s layer: 

n . 1 1 ^ • 11 1 . • the part or 

spheric clouds Hoat m, and under, an atmosphere contaming the solar 
a considerable quantity of the same vapors out of which they atmosphere 
themselves have been formed. This vapor-laden atmosphere, enveiopino- 
probably comparatively shallow, constitutes the reversing lager, the photo- 
By its general absorption it produces the peculiar darkening at ^{^^^"^ 
the limb of the sun, and by its selective absorption it produces 
the dark Fraunhofer lines or solar spectrum. It will be remem- 
bered that Sir Norman Lockyer and others have been disposed 
to question the existence of any such shallow absorbing stratum, 
but that the photographs made at the recent eclipses seem to 
establish its reality. 

The chromosphere and prominences are chiefly composed of The chromo- 
permanent gases, mainly hydrogen, helium, and calcium, which, ^P^^^re and 

^ .y o ^ promi- 

near the photosphere, are mingled with the vapors of the revers- 


ing stratum, but rise to far greater elevations than those vapors, composed 
The appearances are for the most part as if the chromosphere permanent 
were formed of jets of heated gases, ascending "through the inter- gases, 
spaces between the photospheric clouds, like flames playing over 
a coal fire. 

280. The Corona also rests at its base on the photosphere. The corona 
and the characteristic green line of its spectrum is brightest iust ^^^^^ *° ^ 

. . . -1 • 1 great extent 

at the surface of the photosphere, m the reversing stratum, and a mystery. 

in the chromosphere itself ; but it extends far beyond even the 

loftiest prominences to distances sometimes of several millions 

of miles. It seems to be not entirely gaseous, but to contain, 

in addition to the mysterious coronium, dust and fog of some 

sort, very likely of meteoric origin. Many of the phenomena 

of the corona are still unexplained, and since thus far it has 



been observable only during the brief moments of solar eclipses, 
progress in its study has been necessarily slow. No observer 
has yet seen the corona for a sum total of time amounting to 
fifteen minutes. 

Fig. 109 (from The Sun, by permission of D. Apple ton 
& Co.) presents the theory stated above, though the distinction 

Fig. 109. — Constitution of the Sun 
From The Stm, by permission of the publishers 

between the photospheric cloud shell and the chromosphere is 
hardly brought out as clearly as desirable ; nor is it certain that 
all the spots are cavities, as represented. 

THE SUN 259 


1. Assuming Faye's equation (Sec. 230) for the solar rotation, what are 
the rotation periods at the sun's equator, in solar latitude 30°, in latitude 

' At equator, 25.06 days. 
Lat. 30°, 26.49 " 

4 5°, and at the pole ? 


Lat. 45°, 28.09 " 
I At pole, 31.95 " 

2. Assuming Spoerer's equation (Sec. 230), what would be the results? 

3. AYhat would be the synodic or apparent time of rotation for a spot in 
latitude 45° according to Faye's equation? Ans. 30.43 days. 

4. If the diameter of the sun were doubled, its density remaining 
unchanged, what w^ould be the force of gravity at its surface? 

5. If the sun were expanded into a homogeneous sphere, with a radius 
equal to the distance of the earth from the sun, its mass remaining 
unchanged, what would be the force of gravity at its surface? 

-^ns. leVy of g. 

6. In this case, what change, if any, would result in the orbit of the 
earth? Ans. None. 

7. In the neighborhood of a sun-spot a point is found in its spectrum 
where a portion of the C line (A = 6563.0) is deflected to 6566.0. What 
is the velocity (in the line of sight) of the hydrogen at that point ? (See 
Sec. 255.) Ans. 85.17 miles receding. 

8. How great is the displacement of the hydrogen line F (A. = 4861.5) 
at that point? Ans. 2.22 units (of wave-length). 

9. How great a displacement is produced in the line D (X = 5896.16) 
by a velocity of 100 miles a second? Ans. 3.16 units. 

10. If a luminous body were moving towards us with a A^elocity one 
fourth that of light, what would be the effect upon the apparent length of 
the portion of the spectrum included between two given lines, — say C 
and F? 

11. What if it were moving towards us with the speed of light, and 
what if it were receding at that rate ? 

12. AVhat if the observer were receding with the speed of light, and 
what if he were moving towards it at that rate ? 


13. If the diameter of the sun is decreasing at the rate of 300 feet a 
year, how long before its apparent diameter will have decreased by V V 

A71S. 7927 years. 

14. If the rate of shrinkage be assumed to continue uniform (^.e., 300 
feet a year — an impossible assumption), how long would it be before its 
diameter is diminished by 1%? Ans. Over 150000 years. 

15. How much would its mean density then be increased? 

A71S. About 3%. 

16. Taking the " calory " as equivalent to 428 kilogrammeters of energy, 
what weight falling 100 meters to the surface of the earth would, at the 
end of its fall, possess an energy equal to that of the solar radiation 
received in an hour upon 10 square meters of the earth's surface, admitting 
a loss of 50% absorbed by the air? A7is. 38520 kgm. 

17. Assuming that sunlight at the earth equals 70000 times that of a 
standard candle at a distance of 1 meter, at what distance would the light 
of the sun equal that of a, 2000 candle-power electric arc 10 meters 
distant? Ans. About 59 times the earth's distance. 

18. How does the illumination of a surface by an arc-light of 2000 
candle-power at a distance of 1 meter compare with its illumination by 
sunlight? Ans. Jq. 



Form and Dimensions of Sliadows — Eclipses of the Moon — Solar Eclipses — 
Total, Annular, and Partial — Ecliptic Limits and Number of Eclipses in a Year 
— Recurrence of Eclipses and the Saros — Occultations 

281. The Avord "eclipse" is a term applied to the darkening 
of a heavenly body, especially of the sun or moon, though some 
of the satellites of other planets besides the earth are also 

*•' eclipsed." An eclipse of the moon is caused by its passage Eclipses 
throup^h the shadow of the earth; eclipses of the snn^ by the caused by 

^ . , shadows. 

interposition of the moon between the sun and the observer, or, 
what comes to the same thing, by the passage of the moon's 
shadow over the observer. 

The shadow which causes an eclipse is the space from which Shadows of 
sunlight is excluded by an intervening body: geometricallv ^^^*^^^*^ 
speaking, it is a solid^ not a surface. If we regard the sun and cones, 
the other heavenly bodies as spherical, these shadows are co7ies 
with their axes in the line joining the centers of the sun and 
the shadow-casting body, the point being always directed away 
from the sun. 

282. Dimensions of the Earth's Shadow. — The length of the Dimensions 

earth's shadow is easily found. In Fig. 110 we have, from 

•^ ^ shadow of 

of the 
shadow. „ 

the similar triangles, OED and ECa^ the earth. 

OD'.OE'.'.Ea: EC, or L, 

OD is the difference between the radii of the sun and the earth, 
= R — 7\ Ea = r, and OE is the distance of the earth from the 
sun = D. Hence, / \ i 




Length of 
857000 miles. 

(The fraction 108.5 is found by simply substituting for R and 
r their values, R being 109.5 x r.) This gives 857000 miles 
for the length of the earth's shadow when D has its mean value 
of 93 000000 miles. The length varies about 14000 miles on 
each side of the mean, in consequence of the variation of the 
earth's distance from the sun at different times of the year. 

From the cone aCh all sunlight is excluded, or would be were 
it not for the fact that the atmosphere of the earth by its refrac- 
tion bends some of the rays into this shadow. The effect of 

Fig. 110. — The Earth's Shadow 



this atmospheric refraction is to increase the diameter of the 
shadow about two per cent where the moon crosses it, but to 
make it less perfectly dark. 

283. Penumbra. — If we draw the lines Ba and Ah (Fig. 110), 
crossing at P, between the earth and the sun, they will bound 
the penumbra^ within which a part, but not the whole, of the 
sunlight is cut off ; an observer outside of the shadow but 
within this cone frustum, which tapers towards the sun, would 
see the earth as a black body encroaching on the sun's disk. 

While the boundaries of the shadow and penumbra are per- 
fectly definite geometrieally^ they are not so optically. If a 
screen were placed at Jf, perpendicular to the axis of the shadow, 
caiiyindefi- j^q sharply defined lines would mark the boundaries of either 
geometrically shadow or penumbra. Near the edge of the shadow the penum- 
definite. "bra would be very nearly as dark as the shadow itself, only a 

of shadow 
and pe- 
numbra opti 


mere speck of the sun being there visible; and at the outer 
edge of the penumbra the sliading would be still more gradual. 

284. Eclipses of the Moon. — The axis of the central line of 
the earth's shadow is always directed to a point directly opposite 
the sun. If, then, at the time of the full moon, the moon 
happens to be near the ecliptic (z.e., not far from one of the Why 
nodes of her orbit), she will pass throug^h the shadow and be ®^ ^^^®^ , 

^^ ^ c> Qi moon CIO 

eclipsed. Since, however, the moon's orbit is inclined to the not occur 
ecliptic at an average angle of 5° 8', lunar eclipses do not ^* every 
happen very frequently, — seldom more than twice a year. 
Ordinarily the full moon passes north or south of the shadow 
without touching it. 

Lunar eclipses are of two kinds, — partial and total : total Partial and 
when she passes completely into the shadow, partial when she ^ f, ^^ ^^^^'^ 

I: ir J •> r of the moon. 

only partly enters it, going so far to the north or south of the 
center of the shadow that only a portion of her disk is obscured. 

285. Size of the Earth's Shadow at the Point where the Moon Diameter 
crosses it. — Since JEC, in Fig. 110, is 857000 miles, and the ^^ earth's 

^ . shadow 

distance of the moon from the earth is on the average about where the 
239000 miles, CM must average 618000 miles, so that MN, ^^^^n 
the semidiameter of the shadow at this point, will be |^| of the 
earth's radius. This gives ifi\^= 2854 miles, and makes the 
whole diameter of the shadow a little over 5700 miles, — about 
two and two-thirds times the diameter of the moon. But 
this quantity varies considerably with the moon's distance ; the 
shadow, where she crosses it, is sometimes more than three 
times her diameter, sometimes hardly more than twice. 

An eclipse of the moon, when central, i.e., when the moon 
crosses the center of the shadow, may continue total for about 
two hours, the interval from the first contact to the last being Duration of 
about two hours more. This depends upon the fact that the ^ 5V"^^" 

^ /- ^ eclipse. 

moon's hourly motion is nearly equal to its own diameter. 

The duration of a non-central eclipse varies, of course, 
according to the part of the shadow traversed by the moon. 



limits : 
9° 30' to 
12° 15^ 

of a lunar 

286. Lunar Ecliptic Limit. — The lunar ecliptic limit is the 
greatest distance from the node of the moon's orbit at which 
the sun can be at the time of a lunar eclipse. This limit 
depends upon the inclination of the moon's orbit, which is 
somewhat variable, and also upon the distance of the moon 
from the earth at the time of the eclipse, which is still more vari- 
able. Hence, we recognize two limits, — the major and minor. 

If the distance of the sun from the node at the time of full 
moon exceeds the major limit, an eclipse is impossible ; if it is 
less than the minor, an eclipse is inevitable. The major limit 
is found to be 12° 15'; the minor, 9° 30'. 

Since the sun, in its annual motion along the ecliptic, travels 
12° 15' in less than thirteen days, it follows that every eclipse 
of the moon must take place within thirteen days from the time 
when the sun crosses the node. 

287. Phenomena of a Total Lunar Eclipse. — Half an hour or 
so before the moon readies the shadow, its limb begins to be 

Fig. 111. — Light hent into Earth's Shadow hy Refraction 

sensibly darkened by the penumbra, and the edge of the shadow 
itself when it first attacks the moon appears nearly black by 
contrast with the bright parts of the moon's surface. To the 
naked eye the outline of the shadow looks reasonably sharp; 
but even with a small telescope it is found to be indefinite, and 
with a large telescope and high magnifying power it becomes 
entirely indistinguishable, so that it is impossible to determine 
Moment of withiii about half a minute the time when the boundary of 
begmmng not ^j^^ shadow reaches any particular point on the moon. After 

accurately ^ ^ ^ 

observable, the moon lias wholly entered the shadow her disk is usually 


distinctly visible, illuminated Avith a dull, copper-colored light, 
which is sunlight, deflected around the earth into the shadow 
by the refraction of our atmosphere, as illustrated by Fig. 111. 

Even when the moon is exactly central in the largest possible shadow, 

an observer on the moon wonld see the disk of the earth surrounded by a 

narrow ring of brilliant light, colored with sunset hues by the same vapors Color of 

which tinge terrestrial sunsets, but acting with double power because the ^^^^ eclipsed 

light has traversed a double thickness of our air. If the weather happens ^„ 

• fi • J- ^ Illumination 

to be clear at this portion of the earth (upon its rim, as seen from the caused bv 

moon), the quantity of light transmitted through our atmosphere is very light de- 
considerable, and the moon is strongly illuminated. If, on the other hand, fleeted into 
the weather happens to be stormy in this region of the earth, the clouds cut 
off nearly all the light. In the lunar eclipse of 1884 the moon was abso- , 
lutely invisible for a time to the naked eye, — a very unusual circumstance 
on such an occasion. 

the shadow 
by our 

The heat radiation of the moon, according to the observa- 
tions of Lord Rosse, falls off during the progress of the eclipse, 
almost in the same ratio with the light. At the moment when 
the eclipse becomes total fully ninety-eight per cent of the heat Effect upon 
has disappeared, and half of the remaining two per cent is lost ^^^^ mf^on's 
during the totality. As the light returns the heat rises almost ation. 
as rapidly as it fell, showing that the moon's surface has very 
little power of storing heat, — a natural consequence of its 
airlessness; but it is several hours before the heat radiation 
recovers fully the value it had before the eclipse. 

If the eclipse is well visible in both hemispheres, arrange- 
ments are usually made to observe as many star occultations as. 
possible during the totality, for the purpose of determining the 
moon's place and parallax, and for other purposes also. 

288. Computation of a Lunar Eclipse. — Since all the phases why the 
of a lunar eclipse are seen everywhere at the same absolute calculation 

■^ of a lunar 

instant wherever the moon is above the horizon, it follows eclipse is 
that a single computation giving the Greenwich times of the sinipie. 
different phenomena is all that is needed. Such computations 
are made and published in the Nautical Almanac. Each 



observer has only to correct the predicted time by simply 
adding or subtracting his longitude from Greenwich, in order 
to get the true local time. The computation of a lunar eclipse 
is not at all complicated. 

For the method of projecting and computing a lunar eclipse, see Appen- 
dix, Sees. 703 and 704. 

Length of 
the moon's 

diameter of 
of moon's 
shadow on 
the eartli 
168 miles. 


289. Dimensions of the Moon's Shadow. — By the same 
method as that used for the shadow of the earth (Sec. 282) we 
find that the length of the moon's shadow at any time is very 

The Moon's Shadow on the Earth 

^^-^ of its distance from the sun, and averages 232150 


miles. It varies not quite 4000 miles each way, ranging from 

228300 to 236050 miles. 

Since the mean length of the shadow is less than the mean 
distance of the moon from the earth (238800 miles), it is evi- 
dent that on the average the shadow will not reach the earth. 

On account of the eccentricity of the moon's orbit, she is 
much of the time considerably nearer than at others and may 
come within 221600 miles from the earth's center, or about 
217650 miles from its surface. If at the same time the shadow 
happens to have its greatest possible length, its point may reach 
nearly 18400 miles beyond the earth's surface. In this case 
the cross-section of the shadow where the earth's surface cuts 
it squarely (at o in Fig. 112) will be about 168 miles in diam- 
eter^ which is the largest value possible. If, however, the shadow 
strikes the earth's surface obliquely, the shadow spot will be 


oval instead of circular, and the extreme length of the oval ma;^ 
much exceed the 168 miles. 

Since the distance of the moon may be as great as 252970 miles 
from the earth's center, or nearly 249000 miles from its surface, 
while the shadow may be as short as 228300 miles, we may have 
the state of things indicated by placing the earth at B in Fig. 112. Maximum 
The vertex of the shadow, V, will then fall 21000 miles short of diameter 

of the 

the surface, and the cross-section of the shadow produced will shadow 

have a diameter of 196 miles at o\ where the earth's surface Produced 
cuts it. When the shadow falls near the edge of the earth the 
breadth of this cross-section may be as great as 230 miles. 

290. Total and Annular Eclipses. — To an observer within Total and 
the true shadow cone (i.e., between Fand the moon in Fi^. 112) ^°_""^^^ 

^ . eclipses. 

the sun will be totally eclipsed. An observer in the "pro- 
duced" cone beyond F will see the moon smaller than the sun, 
leaving an uneclipsed ring around it, and will have what is 
called an annular^ or " ring-formed," eclipse. These annular 
eclipses are considerably more frequent than the total, and now 
and then an eclipse is annular in part of its course across the 
earth and total in part. (The point of the moon's shadow 
extends in this case beyond the nearest part of the surface of 
the earth, but does not reach as far as its center.) 

291. The Penumbra and Partial Eclipses. — The penumbra can 
easily be shown to have a diameter on the line CD (Fig. 112) width of 
of a trifle more than twice the moon's diameter. An observer *^® ^^^*^ °^ 


situated within the penumbra has a partial eclipse. If he is eclipse on 
near the cone of the shadow, the sun will be mostly covered ®^^^^ ^^^® °^ 

the central 

by the moon; but if near the outer edge of the penumbra, the ime. 
moon will only slightly encroach on the sun's disk. While, 
therefore, total and annular eclipses are visible as such only by 
an observer within the narrow path traversed by the shadow 
spot, the same eclipse will be visible as a partial one every- 
where within 2000 miles on each side of the path. The 2000 
miles is to be reckoned perpendicularly to the axis of the 



Velocity of 
the moon's 
shadow over 
the earth's 

duration of 
a total solar 
eclipse T^SS^. 

for an 

Solar ecliptic 

shadow, and may correspond to a much greater distance on the 
spherical surface of the earth. 

292. Velocity of the Shadow and Duration of Eclipses. — Were 
it not for the earth's rotation, the moon's shadow would pass an 
observer at the rate of nearly 2100 miles an hour on the average. 
The earth, however, is rotating towards the east in the same 
general direction as that in which the shadow moves, and at 
the equator its surface moves at the rate of about 1040 miles 
an hour. An observer, therefore, on the earth's equator with 
the moon at its mean distance from the earth and near the 
zenith would, on the average, be passed by the shadow with a 
speed of about 1060 miles an hour (2100 — 1040), — about 
equal to that of a cannon-ball. In higher latitudes, where the 
surface velocity due to the earth's rotation is less, the relative 
speed of the shadow is higher ; and where the shadow falls very 
obliquely, as it does wlien an eclipse occurs near sunrise or 
sunset, the advance of the shadow on the earth's surface may 
become very swift, — as great as 4000 or 6000 miles an hour. 

A total eclipse of the sun observed at a station near the 
equator, under the most favorable conditions possible, may con- 
tinue total for 7 ""5 8^. In latitude 40° the duration can barely 
equal 6i™. The greatest possible excess of the apparent 
semidiameter of the moon over that of the sun is only 1' 19''. 

At the equator an annular eclipse may last for 12"^24% the 
maximum width of the ring of the sun visible around the moon 
being 1' 37''. 

In the observation of an eclipse four contacts are recognized : the first 
when the edge of the moon first touches the edge of the sun, the second 
when the eclipse becomes total or annular, the third at the cessation of the 
total or annular phase, and the fourth when the moon finally leaves the 
solar disk. From the first contact to the fourth the time may be a little 
over four hours. 

293. The Solar Ecliptic Limits. — It is necessary, in order to 
have an eclipse of the sun, that the moon should encroach on 



the cone ACBI) (Fig. 113), which envelops the earth and sun. 

In this case the true angular distance between the centers of 

the sun and moon, i.e.^ their distance as seen from the center of 

the earth, would be the angle MES} This angle may range 

from 1" 34' 13'' to 1° 24' 19", according to the changing dis- Limits for 

tance of the sun and moon from the earth. The corresponding- P^^'^^^ 

^ ° eclipse 

distances of the sun from the node, taking into account also 15° 2r to 
the variations in the inclination of the moon's orbit, give ^^°'^^'- 
18° 31' and 15° 21' for the major and minor ecliptic limits. 

In order that an eclipse may be central (total or annular) at Limits for 
any part of the earth, it is necessary that the moon should lie ^®^.*^'^ 

? to 11° 50'. 

Fig. 113. — Solar Ecliptic Limits 

wholly inside the cone ACBD^ as M\ and the corresponding 
major and minor central ecliptic limits come out 11° 50' and 
9° b^'. 

294. Phenomena of a Solar Eclipse. — There is nothing of Phenomena 
special interest until the sun is nearly covered, thous^h before ^^^^^^^i 

, . Ill TIP- • ^ solar eclipse. 

that time the shadows cast by the foliage begin to be peculiar. 

The light shining through every small interstice among the leaves, 
instead of forming as usual a circle on the ground, makes a little crescent, 
— an image of the partly covered sun. 

"^MES equals the sun's angular semidiameter SEA + the moon's semidiam- 
eter MEF + the angle AEF ; and AEF equals the difference between EFC^ 
the moon's parallax, and CAE, the parallax of the sun ; hence, as usually 
written, MES, the "radius of the shadow," = S + S' + P — p, P being the 
parallax of the moon, and p that of the sun. 



Effect on 
color of sun- 
light just 
before and 

Advance of 
the shadow, 
and the 

of the corona 
and promi- 
nences and 
of the stars. 

not very 

to be made. 

About ten minutes before totality the darkness begins to be 
felt, and the remaining light, coming, as it does, from the edge 
of the sun alone, is much altered in quality, being very deficient 
in the hlue and violet^ so that it produces an effect very like 
that of a calcium light rather than sunshine. Animals are 
perplexed, and birds go to roost. The temperature falls, and 
sometimes dew appears. In a fcAV moments, if the observer is 
so situated that his view commands the distant horizon, the 
moon's shadow is seen coming, much like a heavy thunder- 
storm, and advancing with almost terrifying swiftness. Just 
before the shadow reaches the observer, quivering, ripple-like 
bands appear on every white surface; and immediately on its 
arrival, and sometimes a little before, the corona and promi- 
nences become visible, while the brighter planets and the stars 
of the first two or three magnitudes make their appearance. 
The suddenness with which the darkness falls is startling. The 
sun is so brilliant that even the small portion which remains 
visible up to within a very few seconds of the total obscuration 
so dazzles the eye that it is unprepared for the sudden transition. 
In a few moments, however, vision adjusts itself, and it is then 
found that the darkness is not really very intense. 

If the totality is of short duration (that is, if the diameter of 
the moon exceeds that of the sun by less than a minute of arc), 
the corona and chromosphere, the lower parts of which are very 
brilliant, give a light at least three or four times that of the 
full moon. Since, moreover, in such a case the shadow is of 
small diameter, a large quantity of light is also sent in from the 
surrounding air, where, 30 or 40 miles away, the sun is still 
shining. In such an eclipse there is not much difficulty in 
reading an ordinary watch face. In an eclipse of long duration, 
say five or six minutes, it is much darker, and lanterns become 

295. Observation of an Eclipse. — A total solar eclipse offers 
opportunities for numerous observations of great importance 


which are possible at no other time, besides certain others which 
can also be made during a partial eclipse. We mention: 

{a) Times of the four contacts, and direction of the line join- 
ing the '' cusps " of the partially eclipsed sun. These observa- 
tions determine with extreme accuracy the relative positions of 
the sun and moon at the moment, (h) The search for intra- 
mercurial planets. (<?) Observations of certain peculiar dark 
fringes, the so-called " shadow bands," which appear upon the 
surface of the earth for about a minute before and after totality, 
(c?) Photometric measurement of the intensity of light at dif- 
ferent stages of the eclipse, (e) Telescopic observations of the 
details of the prominences and of the corona. (/) Spectroscopic 
observations (both visual and photographic), upon the "flash 
spectrum " and upon the spectra of the lower atmosphere of 
the sun, of the prominences and of the corona, [g) Observa- 
tions with the polariscope upon the polarization of the light 
of the corona, (li) Drawings and photographic pictures of the 
corona and prominences. (^) Miscellaneous observations upon 
meteorological changes during the progress of the eclipse, — 
barometer, thermometer, wind, etc., — and effects upon the 
magnetic elements. 

296. Calculation of a Solar Eclipse. — The calculation of a The caicuia- 
solar eclipse cannot be dealt with in any such summary way as *^^^ °^ ^ 

. solar eclipse 

that of a lunar eclipse, because the times of contact and other much more 
phenomena are different at every different station. Moreover, laborious 

. p , . 1 • p 1 1 than that of 

smce tne phenomena oi a solar eclipse admit oi extremely a lunar 

accurate observation, it is necessary to take account of numer- because the 

ous little details which are of no importance in lunar eclipses, stances de- 

The Nautical Almanacs give, three years in advance, a chart of pend on the 

the track of every solar eclipse, and with it data for the accurate ^!^^® °^ ^^^ 
•^ ^ ' observer. 

calculation of the phenomena at any given place. 

T. Oppolzer, a Viennese astronomer, no longer living, published a few 
years ago a remarkable book, entitled The Canon of Eclipses, containing 
the elements of all eclipses (8000 solar and 5200 lunar) occurring between 



Number of 
eclipses in 
a year. 

The eclipse 
mouths and 
the eclipse 

eclipses in a 
year range 
from none 
to three. 

the year 1207 B.C. and a.d. 2162, with maps showing the approximate 
track of the moon's shadow on the earth. It indicates total eclipses visi- 
ble in the United States in 1918, 1923, 1925, 1945, 1979, 1984, and 1994. 

297. Number of Eclipses in a Year — The least possible num- 
ber is two^ both of the sun ; the largest seven^ five solar and two 
lunar or four solar and three lunar. The most usual number of 
eclipses is four. 

The eclipses of a given year always take place at two opposite 
seasons (which may be called the eclipse months of the year), 
near the times when the sun crosses the nodes of the moon's 
orbit. Since the nodes move westward around the ecliptic 
once in about nineteen years (Sec. 192), the time occupied by 
the sun in passing from a node to the same node again is only 
346.62 days, which is sometimes called the eclipse year. 

In an eclipse year there can be but two lunar eclipses, since 
twice the maximum lunar ecliptic limit (2 x 12° 15') is less 
than 29° 6', the distance the sun moves along the ecliptic in 
a synodic month; the sun therefore cannot possibly be near 
enough the node at both of two successive full moons ; on the 
other hand, it is possible for a year to pass without any lunar 
eclipse, the sun being too far from the node at all four of the 
full moons which occur nearest to the time of its node passage. 

In a calendar year (of 365i days) it is, however, possible to 
have three lunar eclipses. If one of the moon's nodes is passed 
by the sun in January, it will be reached again in December, 
the other node having been passed in the latter part of June, 
and there may be a lunar eclipse at or near each of these three 
node passages. This actually occurred in 1852 and 1898, and 
will happen again in 1917. 

As to solar eclipses, it is sufficient to say that the solar ecliptic 
limits are so much larger than the lunar that there must he at 
least one solar eclipse at each node passage of the year, at the 
new moon next before or next after it ; and there may be two^ 
one before and one after, thus making four in the eclipse year. 


(When there are two solar eclipses at the same node, there will Solar 
always be a lunar eclipse at the full moon between them.) In ®^ ^^^^^ 

"J ^ ' range from 

the calendar year a fifth solar eclipse may come in if the first two to five, 
eclipse month falls in January. Since a year with five %olar Greatest 
eclipses in it is sure to have two lunar eclipses in addition, possible 

^ . . , number of 

they will make up Beven in the calendar year. This will eclipses in a 
happen next in 1935; but in 1917 there will also be seven year seven; 
eclipses, — four of the sun and three of the moon. ^^^ two 

298. Frequency of Solar and Lunar Eclipses. — Taking the both of 
whole earth into account, the solar eclipses are the more numer- 
ous, nearly in the ratio of three to two. It is not so, however, 

with those which are visible at a given place. A solar eclipse can 

be seen only from a limited portion of the globe, while a lunar Relative 

eclipse is visible over considerably more than half the earth, — frequency 

. . . . of solar 

either at the beginning or the end, if not throughout its whole and lunar 
duration; and this more than reverses the proportion between eclipses. 
lunar and solar eclipses for any given station. 

Solar eclipses that are total somewhere or other on the earth's 
surface are not very rare, averaging one for about every year Rareness of 
and a half. But at any given place the case is very different ; *°*^^ ^°^^^ 

, -^ . eclipses at 

since the track of a solar eclipse is a very narrow path over the any given 
earth's surface, averaging only 60 or 70 miles in width, we find station, 
that in the long run a total eclipse happens at any given station 
only once in about 360 years. 

During the nineteenth century seven shadow tracks traversed 
the United States, and there will be the same number in the 

299. Recurrence of Eclipses ; the Saros. — It was known to 
the Chaldeans, even in prehistoric times, that eclipses occur 
at a regular interval of 18^11^'^ (10-J- days, if there happen to be 
five leap-years in the interval). They named this period the 

Saros. It consists of 223 synodic months, containing 6585.32 The Saros. 
days, while 19 eclipse years contain 6585.78. The difference 

1 This does not take into account our insular possessions. 



Number of 
eclipses in 
one Saros. 

Star occulta- 

of the dis- 
and reap- 
pearance of 
the star. 

is only about 11 hours, in which time the sun moves on the 
ecliptic about 28'. 

If, therefore, a solar eclipse should occur to-day with the sun 
exactly at one of the moon's nodes, at the end of 223 months the 
new moon will find the sun again close to the node (28' west of 
it), and a very similar eclipse will occur again ; but the track of 
this new eclipse will lie about 8 hours of longitude further west 
on the earth, because the 223 months exceed the even 6585 days 
by y^^2_ of a day. The usual number of eclipses in a Saros is 
about seventy-one, varying two or three one way or the other. 

300. Occultations of Stars. — In theory and computation 
the occultation of a star is identical with a total solar eclipse, 
except that the shadow of the moon cast by the star is sensibly 
a cylinder instead of a cone, and has no penumbra. Since the 
moon always moves eastward, the star disappears at the moon's 
eastern limb, and reappears on the western. Under all ordinary 
circumstances both disappearance and reappearance are instan- 
taneous, indicating not only that the moon has no sensible 
atmosphere, but also that the (angular) diameter of even a very 
bright star is less than 0".02. Observations of occultations 
determine the place of the moon in the sky with great accuracy, 
and when made at a number of widely separated stations they 
furnish a very precise determination of the moon's parallax and 
also of the difference of longitude between the stations. 

observed at 

Occasionally the star, instead of disappearing suddenly when struck by 
the moon's limb (faintly visible by "earth-shine"), appears to cling to the 
limb for a second or two before vanishing. In a few instances it has been 
reported as having reappeared and disappeared a second time, as if it had 
been for a moment visible through a rift in the moon's crust. In some 
cases the anomalous phenomena have been explained by the subsequent 
discovery that the star was double, but many of them still remain mysteri- 
ous ; it is quite likely that they were often illusions due to physiological 
causes in the observer. 


The Laws of Central Force — Circular Motion — Kepler's Laws, and Newton's 
Verification of the Theory of Gravitation — The Conic Sections — The Problem 
of Two Bodies — The Parabolic Velocity — Exercises — The Problem of Three 
Bodies and Perturbations — The Tides 

It is out of the question to attempt here an extended treat- 
ment of the theory of the motions of the heavenly bodies, but 
there are certain fundamental facts and principles easily under- 
stood and so important, and indeed essential, to an intelligent 
comprehension of the mechanism of the solar system that we 
cannot pass them without notice. 

301. Motion of a Body not acted upon by Any Force — Accord- Motion of 
ing to the first law of motion, a moving body left to itself describes ^^^^ °°^ 

^ . . , , J iJ ^ ^ acted on 

a straight line with a uniform speed. When, therefore, we find by force. 
a body so moving we may infer that it is acted on by no force 
whatever or, at least, that if any forces are acting, they exactly 
balance each other, their resultant being zero, and absolutely 
without effect upon the motion of the body. 

It is a common blunder to speak of such a body as actuated 
by a "projectile force," — a survival of the Aristotelian idea Nopro- 
that rest is more natural to a bodv than motion, and that J®^*^ ® 

^ force 

"force" must operate to keep a body moving. This is not true: required to 
mere motion implies no acting force. Change of motion only, 
either in speed or in direction, implies such action. 

With the notion referred to there usually goes another, — that a moving 
body must have been put in motion by some force, as if all bodies were 
once at rest — say at the moment of creation — and acquired their motion 
later ; in respect to which we have no knowledge. 


free motion. 



Motion of 
body under 
force acting 
in the line 
of motion. 

under force 
acting across 
line of 

Condition of 



Only a 
single force 
needed to 

Law of 
under a 

tion that an 

302. Motion under the Action of a Force. — If the motion of 
a body is in a straight line but with a varying speed, we infer a 
force acting directly in the line of motion, either accelerating 
or retarding. If the body a moves in a curve (Fig. 114), we 
know that some force is acting crosswise to the motion and 
towards the concave side of the curve. If the speed increases, 
we know that the acting force pulls not only crosswise, but 
forward, as «5, making an angle of less than 90° with the "line 
of motion," at^ tangent to the curve at a; and vice versa if the 

motion is retarded. 

If the speed keeps constant, we 

know that at a the force acts along 

ac^ always exactly perpendicular to 

the line of motion. 

Fig. 114. — Curvature of an Orbit 

It is not unusual to find curvilinear 
motion spoken of as due necessarily to 
two forces; one, the "projectile force," 
imagined to act along the line of motion, v^^hile the second force draws 
sidewise. There may have been a projectile force acting in the past, but 
if so it is "ancient history"; we need, at present, in order to explain the 
facts, the action of only a single force, operating to change the direction 
or the speed, or both, of the body's motion. From a cm^ved path we can 
infer the necessary existence of hut one force. This force may be, and 
often is, the "resultant" of several; but then they act as one, and only 
one is needed. 

303. Laws governing the Motion of a Body moving under the 
Action of a Force directed to a Fixed Center ; Law of Areas. — 
In this case it is obvious that the path of the body will be a 
curve, concave towards the center of force, and all lying in one 
plane with that center. 

It is easy to prove, further, that it will move in such a way 
that its radius vector will describe equal areas in equal times 
around that point. 

Imagine a body moving uniformly along the straight line 
AB (Fig. 115), so that AB, BC, CL (the spaces described in 



successive seconds) are all equal; then, wherever may be, the 
triangles AOB, BOC, COL, etc., are all equal, having equal 
bases and the common vertex 0. A body in uniform rectilinear 
motion therefore describes with its radius vector equal areas in 
equal times around any point whatever. 

Suppose, now, that when the body reaches C a blow or 
impulse directed towards is given, imparting a velocity which 
would carry it to K in one second if it had been at rest when 
struck. The resultant of the original motion CL, combined 
with the newly imparted motion CK, is CD, found according to 
the "parallelogram of velocities" {Physics^ p. 18) by drawing 
KD and LD parallel, respectively, to CL and CK, so that at the 
end of a second the body will 
arrive at D instead of going to 
i, and its velocity will have 
become CD instead of BC. 

Now the area of the tri- 
angle COD equals that of 
COL, because they have the 
common base (70, and their 
vertices are on a line, i>X, 
parallel to that base, making 
their "altitude" the same. 
But COL = BOC\ therefore 
COD= COB, It follows, 
therefore, that when a moving body receives an impulse directed 
towards a given point the area described by the radius vector in a 
second around that point remains unchanged by that impulse. 

If the impulse had been directed from the point 0, towards 
K' instead of K, the result would have been the same. The 
same reasoning shows that the area COD^ is equal to COB. 

But if K were not on the radius vector CO, the area would 
be changed, increasing if CK lay between CO and CL, and 
decreasing if between CO and CB. 

towards a 
point does 
not alter 
the area 
described by 
the radius 
that point. 

Fig. 115 



Hence fol- 
lows the 
of uniform 
of areas. 

linear, and 

304. Furthermore, since a continuous force, like attraction, 
directed towards or from a fixed center, 0, may be regarded as 
an uninterrupted succession of little impulses, each directed 
along the radius vector, we have the perfectly general law that 
whenever a hody moves under the sole action of a force directed along 
the radius vector drawn from the hody to a center^ the radius vector 
will describe around that ceyiter areas proportional to the time. It 

makes no difference according to what 
law the intensity of the force varies: 
it may be attractive or repulsive, con- 
tinuous or intermittent, may vary as 
gravity does or with complete irregu- 
larity; but so long as it never acts 
except along the radius vector the 
" areal velocity," as it is called [i.e., 
the number of square feet or acres or 
square miles described by the radius 
vector in a unit of time), remains 
absolutely constant. 

Thus, in Fig. 116, representing part 
of a comet's orbit around the sun, if the arcs ah, cd, ef are each 
described in the same time, then the shaded areas are all equal. 
The converse theorem is also easily proved, viz., that if a 
body moves in a curve in such a way that its radius vector 
drawn to a given point describes equal areas in equal times 
around that point, then the force that acts upon the body is 
always directed to that point. 

305. Areal, Linear, and Angular Velocities. — Areal velocity 
has just been defined. The linear velocity of a body is the 
number of linear units (feet, meters, miles) which it moves 
over in a unit of time, — say a second. Its symbol is usually V. 
The angular velocity is the number of units of angle (radians, 
degrees, seconds) swept over by the radius vector in a unit of 
time. The usual symbol for this is co. 

Fig. 116. — The Law of Equal 


In Fig. 117 if AB is the length of the path described in a 
unit of time, AB is the linear velocity/ V; the angle ASB is the 
angular velocity/, co ; and the area ASB is the areal velocity, which 
is constant. Calling this A and regarding the sector as a tri- 
angle (which it is nearly enough), we have A = ^J^ x j^, p being Formulae 
the line Sb drawn from the center of force perpendicular to the for linear 

^ ^ and angular 

line of motion ; so that if we regard AB as the base of the velocities in 
triangle, ]? is its altitude. Hence, we have the equation ^^^"^'^ °f 

V = — . (1) '''°"'^' 


Also, A = ^ r-^r^ sin ASB. Since in a second of time the 

angle ASB, or &>, is so small that it may be taken equal to its 

sine, and ^^^2 equals (sensibly) r^, we have 

" = ^- (2) 

In every case, therefore, of motion under a central force, 
(1) the areal velocity (square miles per second) is constant in 
all parts of the orbit ; (2) the linear 
velocity (miles per second) varies in- 
versely as p, the perpendicular drawn 
from the center to the line of motion ; 

(3) the angular velocity (radians or ^ \\ \ ^-^ 

degrees per second) varies inversely 
as the square of the radius vector. 

These three statements are not '^ Extreme 

independent laws, but simply differ- Fig. m.- Linear and Angular generality 
, . , • ^ . J- Velocities of the laws. 

ent geometrical equivalents lor one 

law. They hold good regardless of the nature of the force, 
requiring only that when it acts it acts directly towards, or from, 
the center, along the line of the radius vector. 

306. Circular Motion. — In the special case when the path of Central force 

in case of 

to its center, both the linear and angular velocities are constant, motion. 

a body is a circle described under the action of a force directed 

"^ circular 



laws stated. 

as is also the force, which is given by the familiar formula 
already several times used : 



/=4 7r2 



obtained by substituting for V in equation {a) its value, 27rr 
(the circumference of the circle), divided by ^, the time of 
revolution. As the orbits of the principal planets are all 
nearly circular, these formulae will find frequent application. 

307. Kepler's Laws. — Early in the seventeenth century 
Kepler discovered, as unexplained facts, three laws which 
govern the motions of the planets, — laws which still bear his 
name. He worked them out from a discussion of the obser- 
vations which i Tycho Brahe had made through many preced- 
ing years upon the planets, Mars especially. They are as 
follows : 

(1) The orbit of each planet is an ellipse with the sun in one 
of its foci. (See Sec. 160.) 

(2) The radius vector of each planet describes equal areas in 
equal times. 

(3) The squares of the periods of the planets are propor- 
tional to the cubes of their mean distances from the sun; i.e., 

t 2 

t 2 



,^. This is the so-called "Harmonic Law." 

Examples 308. To make sure that the student apprehends the meaning and scope 

Illustrating of this third law, we add a few simple exam]3les of its application : 
the Har- 
monic Law. 1- What would be the period of a planet having a mean distance from 

the sun of one hundred astronomical units, i.e., a distance a hundred times 

that of the earth? 

13 : 1003 = 12(year) : A2 . 

whence. A' (in years) = V 100^ = 1000 years. 

2. What would be the distance from the sun of a planet having a 
period of 1 25 years ? 

12(year) : 125^ 

13 : A3 . whence A = V1252 = 25 astron. units. 


3. What would be the period of a satellite revolving close to the earth's 

surface V 

(moon's dist.)'' : (dist. of satellite)^ = (27.;3 days)'^ : X^, 

or, 603 : 13 = 27.32 :A2; 

whence, X = "^••^_^^^' = Vm^. 

The Harmonic Law as it stands in Sec. 307 is not strictly Modification 
true: it would be so if the planets were mere partieles, iniini- of Harmonic 

•^ _ . Law taking 

tesimal as compared with the sun ; but this is not the case, account of 
though the difference is so slight that Kepler did not detect it. planets' 


Ihe accurate statement, as Newton showed, is 

t^^ (M H- m^) : t^'^ (M + m^) = r^^ : r^^, 

in which M is the sun's mass, and m,-^ and m^ are the masses of 
the two planets compared. In the case of Jupiter the correc- 
tion makes a difference of about two days in its period ; z.e., its 
period is about two days shorter than that of a particle moving 
in the same orbit would be. 

309. For fifty years these laws remained an unexplained 
mystery. Many surmises, partly correct, were early made as 
to their physical meaning. Several persons "guessed" that the 
explanation would be found in a force directed to the sun ; 
Newton proved it. He first demonstrated substantially, as 
given in Sec. 303, the law of equal areas and its converse as 
being in the case of central motion a necessary consequence 
of the three fundamental laws of motion, which he had been Newton 
the first to formulate. He also proved by a demonstration a p^'^^®^ ^^^^^ 

^ f the law of 

little beyond the scope of this book that if a planet moves in gravitation 
an ellipse with the center of force at its focus, then the force follows from 

• • 1 • Kepler's 

acting upon the body at different points m its orbit must vary ^^ws. 
inversely as the square of the radius vector at those points; and, 
finally, he proved that, granting the Harmonic Law, the force 
from planet to planet must also vary according to the same law 
of inverse squares. 



tion for 

laws as to 
the force 
which acts 
on the 

upon mass 
and distance 

310. For circular orbits the proof is very simple. From 
equation (h) (Sec. 306) we have, for the first of two planets, 


/i = 4 


f 2' 

in which /^ is the central force (measured as an acceleration in 
feet per second), and r-^ and t^ are, respectively, the planet's 
distance from the sun and its periodic time. 

For a second planet. 


Dividing the first equation by the second, we get 
But, by Kepler's third law, 

/2./2_^3.., .3 
Cj • '2 — 1 2 


t 2 

whence —- 
1 2 



Substituting this value of -^ in the preceding equation, we have 




^ 2 

J 2 '2 '1 '1 

i.e.^f-^ 1/2 = r^\ r-^^ which is the law of inverse squares. 

In the case of ellijotical orbits the proposition is equally true 
if for r we substitute «, the semi-major axis of the orbit ; but 
the demonstration is much more complicated. 

311. Inferences from Kepler^s Laws. — From Kepler's laws 
we may therefore infer, as Newton proved : First (from the 
law of areas), that the force tvliich determines the orbits of the 
planets is directed towards the sun. 

Second (from the first law), that the force which acts upon any 
given planet varies inversely., at different points in the orhit., as the 
square of the radius vector. 

Third (from the Harmonic Law), that the force tvhich acts 
upon one planet is the same that it would be for any other planet 
put in the place of the first; in other words, the attracting force 


depends only on the mass and distance of the bodies concerned, only, and is 
and is independent of their physical condition^ such as tempera- f®^*^^^^^y 

^ -' r iy •> 1 independent 

ture, chemical constitution, etc. It makes no difference ^et of all other 
detected in the motion of a planet around the sun, whether it circmn- 
is hot or cold, made of hydrogen or of iron; but it would be 
going too far to say that the future may not yet show some 
slight differences depending upon such circumstances. 

312. Newton's Test of his Theory of Gravitation by the Motion Test of the 
of the Moon. — When Newton first conceived the idea of uni- ^^^^^J ^J 


versal gravitation in 1665, he saw at once that the moon's by means of 
motion around the earth ought to furnish a test. Since the ^^^® moon's 

^ _ ^ motion. 

moon's distance (as was well known even then) is about sixty 
times the radius of the earth, the distance it should fall towards 
the earth in a second ought to be, if his idea of gravitation was 

correct, — — , or ^^^^ , of 193 inches (the distance which a body 
602 3600 ^ *^ 

falls in a second at the earth's surface), provided we assume The deiiec- 
that the earth attracts as if its mass were all collected at its ^]^^^^ 

the moon 

center, — to prove which gave Newton much trouble, and towards the 
became possible only after his invention of "fluxions." earth is just 

Now g-gVo" of .193 inches is 0.0535 inches. Does the moon should be 
fall towards the earth, i.e,, deflect from a straight line, by this according to 

, , f. the law of 

amount each second r gravitation. 

According to the law of central forces, considering the moon's 
orbit as circular, /. ^ o ^ 

and the deflection is one half of this, viz., ^tt^t^- ^^ we com- 
pute the result, making r = 238840 miles reduced to inches, 
and t the number of seconds in a sidereal month, the deflection 
comes out 0.0534 inch, a difference of only y-q^q^q^ of an inch, — 
practically a complete accordance. 

Unfortunately for Newton, when he first made this test, the distance of 
the moon in miles was not known, because the size of the earth had not 
then been determined with any accuracy. The length of a degree was 


Why the supposed to be about 60 miles instead of 69, as it really is. Newton com- 

test ap- puted the radius of the earth on this erroneous basis and, multiplying it 

peare o ^ qq obtained for r, the distance of the moon, a quantity about sixteen 
fail when -^ ' ' n j 

Newton first P^^' ^^^^* ^^^ small ; from this he calculated a corresponding deflection of 
applied it. only about 0.04i inch. The discordance between this and 0.0535 was too 
great, and he loyally abandoned the theory as contradicted by facts. 

Six years later, in 1671, Picard's measurement of an arc of the meridian 
in France corrected the error in the size of the earth, and Newton on 
hearing of it at once repeated his calculation, or tried to, for the story goes 
that he was too excited to finish it, and a friend completed it for him. 
The accordance was now satisfactory, and he resumed the subject with 
zeal and soon established the correctness of his theory. 

The test not It IS to be noted that while discordance in even a single case 
sufficient to ^yQuld be fatal to the theory, accordance in a single case does 

demonstrate , , 

the correct- not pi'ove it, but only makes it more or less probable. The 
ness of the demonstration of the law of gravitation lies in its entire accord- 
ance, not with one or two selected facts, but with a countless 
multitude, and in its freedom from a single contradiction shown 
by the most refined observations. 
Its proof Apparent contradictions have now and then cropped out, but 

lies m Its Q^ have found explanation, except, perhaps, one slight diver- 
with all gence at present outstanding (in the motion of the apsides of 
facts the planet Mercury) which thus far baffles the mathematicians, 

ohserved ^^^ vfiW-, in all probability, sooner or later disappear like its 

313. The Inverse Problem. — Newton did not rest with merely 
The inverse sliowing that the motion of the planets and of the moon could 
problem: ^^ explained by the law of gravitation ; but he also investisfated 

to determine l j o ' o^ 

what the and solvcd the more general inverse problem and determined 

orbit must ^^^^ Mnd of motion is necessary according to that law. He 
be if the law . , . '^, , • p . -, . -, i 

is correct. lound that the orbit oi a body moving around a central mass 
under the law of gravitation need not be a circle, nor even an 
ellipse of slight eccentricity like the planetary orbits. But it 
must be a Conic. Whether it will be a circle, ellipse, parabola, 
or hyperbola depends on circumstances. 



314. The Conies. — (1) The ellipse is the section of a cone The two 
made by a plane which cuts completely across it, as EF in ^o^^^s. 
Fig. 118. The ellipse varies in form and size, according to the 
position and inclination of the 
cutting plane, the circle being 
simply a special case when the 
section is perpendicular to the 
axis of the cone. 

(2) The Hyperbola. When 
the cutting plane makes with the 
axis an angle less than BVC 
(the semiangle of the coyie) it 
plunges continually deeper and 
deeper into the cone and never 
comes out on the other side. The 
section in this case is an hyper- 
bola^ GHK. If the cutting plane 
be produced upward, it encoun- 
ters the other nappe of the cone 
(the "cone produced"), cutting 
out from it a second hyperbola, 
G'H'K\ exactly like the first, 
but turned in the opposite direc- 
tion. The pair of twin curves, 
GHK and G'H'K'., are considered 
as two parts of the same hyper- 
bola, the axis of which, HH' in 
the figure, lies between the two 
branches and outside of both, 

and is therefore always reckoned as negative. The center of 
the pair of twin hyperbolas is the middle point of this axis. 

(3) The Parabola. When the cutting plane is parallel to the 
side of the cone^ as FRO, it never cuts in deeper, nor, on the 
other hand, does it run across the cone. The section in this 

Fig, 118. — The Conies 



The parab- 
ola: the 
ellipses and 

case is called Sj parabola, which, so to speak, is the boundary or 
partition between the ellipses and hyperbolas which can be cut 
from a given cone by changing the inclination of a given plane. 
The least deflection of the cutting plane outward from the 
parallel changes the parabola into an ellipse, and into an hyper- 
bola, if inward. 

All parab- All parabolas, of whatever size, and cut from whatever cone, are of the 

olas identi- same shape, as all circles are, — a fact by no means obvious without demon- 

ca m orm, g^^j-^tion, though we cannot give the proof here. This does not mean, 

differing ' 5 & i ' 

only in size. 

Fig. Hi). — The Relation of the Conies to Each Other 

however, that a?i arc of one j)arabola is of the same shape as a7i(/ arc of 
another parabola (taken from a different part of the curve), but that the 
complete parabolas, cut out from infinitely extended cones, are all similar, 
whether the cone be sharp or blunt, or whether the plane cuts it near to or 
far from its vertex. 

The ellipse, 315. The Ellipse, Parabola, and Hyperbola. — Fig. 119 shows 
hyperbola, ^|^g appearance and relation of these curves as drawn upon a 

and parabola , , ... * 

compared as plane. The Ellipse is a " closed curve " returning into itself, 
curves upon ^^^ \^ \^ ^q ^um. of the distances of any point, N, from the 

a plane. . . . . 

two foci equals the major axis ; i.e., FN-\- F'N= PA. 


The Hyperbola does not return into itself, but the two 
branches PN' and Fn" go off into infinity, becoming ultimately 
nearly straight and diverging from each other at a definite angle. 
In the hyperbola the difference of two lines drawn from any 
point on the curve to the two foci equals the major axis; i.e.^ 
F^'N' — FN' — PA\ C'F being the semi-major axis, a, of the 

The Parabola^ like the hyperbola, fails to return into itself, 
but its two branches, instead of diverging, become more and 
more nearly parallel. It has but one accessible focus and may 
be regarded either as an ellipse with its second focus, F\ 
removed to an infinite distance, and therefore having an infinite 
major axis ; or, with equal correctness it may be considered as 
an hyperbola^ of which the second focus, F"^ is pushed indefi- 
nitely far in the opposite direction, so that it has an infinite 
(^negative) major axis. . 

In the ellipse the eccentricity ( -7— ] is less than unity. The eccen- 

\ / / FC"\ tricity of 

In the hyperbola it is greater than unity ( — j j. ellipse is 

\P^ / less than 

In the parabola it is exactly unity ; in the circle, zero. unity; that 

The eccentricity of a conic determines its form. All parab- ^^^yp^^^^i^ 

•^ ^ \B greater 

olas, therefore, are of the same form, as already said, as are all than unity ; 
circles. Of ellipses and hyperbolas there is an infinite variety *^^* "^ 

•^ parabola is 

of forms, from such as are so narrow as to be only a line or a exactly 
pair of diverging lines, to those that are broad as compared with "°i<^y- 
their length. 

316. The Problem of Two Bodies. — This problem, proposed The problem 

of two 

Criven the masses of two spheres and their positions and motions 

and completely solved by Newton, may be thus stated : 

-^ "^ ^ 'J bodies 

. , . . . Motion of 

at any moment ; givefi also the law oj gravitation : required the their com- 

motion of the bodies ever afterwards and the data necessary to "^o^ center 

compute their place at any future time. unaffected 

The mathematical methods by which the problem is solved by their 

require the use of the calculus and must be sought in works on "^^*^^ 

-'' *^ attraction. 


analytical mechanics and theoretical astronomy, but the general 
results are easily understood. 

In the first place, the motion of the center of gravity of the two 
bodies is not in the least affected by their mutual attraction. 
The size of In the next place,, the two bodies will describe as orbits around 
leir or i s ^^\^ commou Center of s^ravity two curves precisely similar in 

inversely o ./ j: a 

proportional form,, but of size inversely proportional to their masses,, the form 
to their ^j^^j dimensions of the two orbits being^ determined by the masses 

masses. . . o ,j 

and velocities of the two bodies. 

If, as is generally the case in the solar system, the two 

bodies differ greatly in mass, it is convenient to ignore the 
The relative Center of gravity entirely and to consider simply the relative 
orbit of the motion of the smaller one around the center of the other. It 

smaller with 

respect to will movc with reference to that point precisely as if its own 
the larger, mass, m, had been added to the principal mass, Jf, while it had 
become itself a mere particle. This relative orbit wiW be pre- 
cisely like the orbit which m actually describes around the 
center of gravity, except that it will be magnified in the ratio 
of {M -\- m) to Ml i.e.,, if the mass of the smaller body is j^-^ of 
the larger one, its relative orbit around M will be just one per 
cent larger than its actual orbit around the common center of 
gravity of the two. 
The relative 317. Finally,, the orbit will always be a "conic," i.e.,, an 
orbit a conic, gUipg^ or afi hyperbola; but which of the two it will be depends 

the species ^ , . it 

and size of 0^ three things, VIZ., the united mass oi the two bodies (M + m), 
which de- ^hc distance, r, between m and M at the initial moment, and the 
the masses velocity, F, of m relative to M. 

their initial If this velocity, F, be less than a certain critical velocity, U, 
and veioci- wMch depends only on (M -\- m) and r and is called the "para- 
ties, bolic velocity" or "velocity from infinity," the orbit will be 
an ellipse; if greater, it will be an hyperbola. If, however, F 
and ?7 should happen to be exactly equal, the orbit would be a 
Criterion for parabola ; but such exact equality is extremely improbable, — 
species. ^YiQ chances are infinity to one against it, 


The direction of the motion of m with respect to M, while it The form 
has influence upon the form of the orbit (its " eccentricity "), has ^^P^"*^^ 

. ... . . . partly upon 

nothing to do with determining its species and semi-major axis direction of 
nor with its period in case the orbit is elliptic ; these are all inde- i^^^^^i^i 

-, PIT • n •> ' motion. 

pendent oi the direction oi m s motion. 

The problem is completely solved. From the necessary initial 
data corresponding to a given moment we can determine the 
position of the two bodies for any instant in the eternal past or 
future, provided only that no force except their mutual attraction 
acts upon them in the time covered hy the calculation. 

318. The Parabolic Velocity. — The parabolic velocity at the Definition 
distance r is also called the "velocity from infinity," because it ^^^^^® 

^ . -^ parabolic 

is the speed which would be acquired by the particle m in falling velocity or 
towards the mass M from an infinite distance to the distance r velocity 
from Jf, — assuming, of course, that M is fixed and that m infinity, 
starts from rest and during its fall is not acted upon by any 
force excepting the attraction between itself and M. It might 
be supposed that this velocity would be infinite, but it is not so 
unless r becomes absolutely zero. It is given by the formula 

-.^ , ^JM-\- m . , _^Im ,^ , Formula 

^r^^''^—;—' ^^ ^^"^"^^y ^\7' (^) for the 

when m is infinitesimal as compared with M. (For a demon- velocity 
stration of this formula the reader is referred to works on ana- 
lytical mechanics.) 

In this formula /c is a constant which depends on the mass of 
ilf .and on the units of measurement employed. If we take the 
mass of the sun as the unit of mass and the radius of the earth as 
the unit of distance for r, it becomes 26.156 miles per second, and 
we have for the parabolic velocity due to the sun's attraction on a 
particle falling from infinity to the distance r, ;j 

U^ (miles per second) = 26.156 \-5 (2) 

and f; 2 = ^84:1^. (2') 


1 Ur signifies " parabolic velocity at distance r." 



velocity at 
surfaces of 
sun, earth, 
and moon. 

between the 
velocity and 
the species - 
of the orbit. 

If the mass of the sun were four times as great, the coeffi- 
cient would be doubled^ since, according to equation (1), ?7 varies 
with the square root of M. At a distance one fourth that of 
the earth from the sun, r would become one fourth and the 
parabolic velocity would also be doubled. At the distance of 
Neptune, where r = 30.05, ?7 is only 4.77 miles per second. 

319. Formula (1) enables us to compute the parabolic velocity at the sur- 
face of any body whose mass and radius are known. In the case of the sun 

M=l and r = — - — (i.e., 433250 -^ 93 000000), so that at the sun's sur- 
214.66 ^ ^ 

face U = 383.2 miles per second ; if a body were ejected from the sun with 

a speed exceeding this, it would go off and never return. 

and r = , . . _^ (Sec. 225), and from equa- 

For the earth, M = 

332000 23467 

tion (1) we find U at the earth's surface equals 6.9 miles per second. 

At the surface of the moon a similar computation gives U as only 
1.48 miles, or less than 8000 feet per second. A body projected from the 
moon with a speed greater than this would never return, and it will be 
recalled that in this fact probably lies the explanation why the moon has 
lost her atmosphere. 

320. Relation between the Parabolic Velocity and the Nature 
of the Orbit of a Body revolving around the Sun. — From theo- 
retical astronomy (Watson, p. 49) we have the equation 






a being the semi-major axis of the orbit of a body, V^ the velocity 
of the body in its orbit at a point whose radius vector is r, and 
yu- a constant which equals 2Mic'^^ — the k of equation (1). From 
equation (1), Mk^ = r X U^% so that fji= 2 r U^^. Substituting this 
value of ^i in equation (3), we find at once 



The semi- 
major axis 
of orbit 

?7,2- F,V 

This equation is of great importance, since it shows that 
the species of the orbit is determined solely by the difference 
between W^ and V^. 


If the denominator of the fraction is positive, the value of a upon the 
will be positive and the orbit will be an ellipse. This is the ,\^^^^^lr^ 

^ ^ ^ ^ between U^ 

case when F^, the orbital velocity at the distance r, is less than and v^ at 
CT., the parabolic velocity at that distance. distance r. 

If, on the other hand, F,. is greater than U^., the denomi- 
nator becomes negative, and so does a, and the orbit is an 

If F,. exactly equals ?7,., the denominator becomes zero, a becomes 
infinite, and the orbit is a parabola. This explains why U is 
called the " parabolic velocity " : at every distance from the 
sun the velocity of a body moving in a parabola is precisely 
what it would have acquired in falling to that point from an 
infinite distance under the sun's attraction. 

If the orbit is an ellipse, the velocity at every point in the 
orbit is less than the parabolic velocity, and greater if the orbit 
is an hyperbola.^ 

In order that a planet may move in a circle around the sun. Condition 
as the principal planets do very nearly, a must equal r, and equa- 
tion (4), by substituting r in place of a, gives 

_ r ?72 ^^ _ o 

whence, F^ = i U'^ and F= f^Vf = 0.7071 x V; i.e., the velocity 
of a body moving in a circular orbit is equal to the parabolic 
velocity multiplied by V|^. 

Vice versa, U= Fv2, and hence the parabolic velocity at 
distance unity (that of the earth from the sun) equals the earth's 

1 The expression for the eccentricity is more complicated than that for the 
semi-major axis, since it involves the angle 7 between the radius vector and the 
tangent drawn at its extremity. The equation is 

The eccentricity is therefore greater than, less than, or equal to unity, according 
as {U^ — V^) is positive or negative. It will be noticed also that no linear 
quantity (r or a) enters into the expression, which determines only the form 
and not the size of the orbit. 

for motion 
in a circle. 



for planet's 

Effect of 
changes of 
upon a 

Behavior of 
the frag- 
ments of an 

orbital velocity, 18.5 miles X V2 = 26.16 miles per second ; and 
this is the way in which the constant k is usually computed. 

321. The Expression for a Planet^s Period. — From theoret- 
ical astronomy (Watson, p. 46) we have the equation 

^ = 27r X ^-^, 




where t is the periodic time. This embodies Kepler's third 
law, and shows that all planets moving in ellipses and having 
the same major axis will have the same period, notwithstanding 
differences in the eccentricity of their orbits. 

Also that if a is infinite, as in the parabola, the period is 
also infinite. 

Also that in the hyperbola (in which a is negative) the period, 
since it involves the square root of the negative quantity a^, is 
imaginary, i.e., in this case impossible. 

When a body is moving in a parabola ( F^ = V^) the least 
decrease of F by a disturbing action will transform the orbit 
into an ellipse with a definite period, or an increase of velocity 
will make it an hyperbola. 

Again, if a planet moving in a circular orbit should have 
its speed increased in a ratio greater than that of the square 
root of 2 to 1, say one and one-half times, it would go off in 
an hyperbolic arc and never return. 

Finally, if a planet were to explode at any point in its orbit, 
all the pieces, except those which had a velocity greater than the 
parabolic velocity at the point of explosion, would move around 
the sun in ellipses, and at every revolution would pass through 
the point where the explosion occurred ; moreover, any frag- 
ments which were thrown off with equal velocities would have 
the same period and after a single circuit around the sun would 
arrive there simultaneously. 



1. Given a comet moving in an ellipse with the eccentricity 0.5. Com- 
pare the velocities, both linear and angular, at the perihelion and aphelion. 

( Lin. vel. at perihelion is three times that at aphelion. 
" ^*'^' "( Ang. vel. " " " nine " " " " 

2. What would be the result if the eccentricity were i? what if it 
w^ere f ? 

3. What would be the periodic time of a small body revolving in a circle 
around the sun close to its surface ? (Apply Kepler's Harmonic Law.) 

Ans. 2H7"».4. 

4. What would be its velocity? Ans. 270.8 miles a second. 

5. If the earth had a satellite with a period of 8 months, what would its 
distance be? Ans. Four times that of the moon. 

6. If Jupiter were reduced to a mere particle, how much would its 
period be lengthened? (Consider its mass to be -^-^^j of the sun's, and 
see Sec. 308.) 

Solution. Let x be the new period ; then, 

x2 : t^ =z r^ : r^ = 1 : 1 , since r is not changed. Whence, 


X = t Jl^ = t{l+ix ToVs + etc.) =t{l + 2 oVe). very nearly. 

/i Q Q O A 

But t = 4332.6 days, and {x - t) = ' ' = 2.0.67 days. Ans. 

7. How much longer would the earth's period be if it were a mere 
particle? Ans. qqj)\q-q of a year, or 47.8 sec. 

8. If the sun's mass were a hundred times greater, what would be the 
parabolic velocity at the earth's distance from it (Sec. 318) ? 

Ans. Ten times its present value, i.e., 261.6 miles a second. 

9. If the sun's mass were reduced 50 per cent, what would be the para- 
bolic velocity at the distance of the earth? Ans. 18.5 miles a second. 

10. If the sun's mass were to be suddenly reduced by 50 per cent or 
more, what would be the effect upon the now practically circular orbits of 
the planets? (See Sec. 320, last paragraph.) 



11. What would be the effect upon the orbit of the earth if the sun's 
mass were suddenly doubled ? 

Ans. It would immediately become an eccentric ellipse, with its 
aphelion near the point where the earth was when the change occurred. 

12. Let F,. be the velocity in an orbit at a point where the radius vector 
is r, and let C/,. and U^^ be the parabolic velocities at distances r and 2 a from 
the sun, a being the semi-major axis of the orbit. Show that 

I ,.■' = U,. ± L 2a • 

The plus sign applies if the orbit is an hyperbola, the minus sign if it is an 

In words this may be stated thus (siuce the energy of a moving body is 
proportional to the square of its velocity) : 

The energy of a body moving in an orbit under gravitation, when at a distance 
r from the center of attraction, equals the energy it would have acquired by falling 
to r from infinity ±_ the energy it would have acquired by falling from infinity 
to the distance 2 a, the major axis of the orbit. 

of the 
problem of 
three bodies. 

The general 
but special 
cases solved. 



322. As has been said, the problem of two bodies is completely 
solved ; but if instead of two spheres attracting each other we 
have three or more, the general problem of determining their 
motions and predicting their positions transcends the present 
power of human mathematics. 

"The problem of three bodies" is in itself as determinate and 
capable of solution as that of two. Given the initial data, i.e.^ 
the masses^ positions^ and motions of the three bodies at a given 
instant; then, assuming the law of gravitation, their motions 
for all the future and the positions they will occupy at any 
given date are absolutely predetermined. The difficulty is with 
our mathematics. 

But while the general problem of three bodies is intractable, 
nearly all the particular cases of it which arise in the considera- 
tion of the motions of the moon and of the planets have already 
been practically solved by special devices, Newton himself 
leading the way; and the strongest proof of the truth of the 


theory of gravitation lies in the fact that it not only accounts 
for the regular elliptic motions of the heavenly bodies, but 
also for their apparent irregularities. 

323. It is quite beyond the scope of this work to discuss the 
methods by which we can determine the so-called " disturbing 
forces" and the effects they produce upon the otherwise elliptical 
motion of the moon or of a planet. We make only two or three 

Firsts that the " disturbing force " of a third body upon two Disturbing 
which are revolving around their common center of p-ravity is *°^^® ^®" 

^^ _ . pends upon 

not the whole attraction of the third body upon either of the two, the differ- 
but is generally only a small component of that attraction. It ®^^® ^^ 

. attractions 

depends upon the difference oj the two attractions exerted by the upon the 
third body upon each of the pair whose relative motions it disturbs., bodies dis- 
— a difference either in intensity^ or in direction., or in both. 

If, for instance, the sun attracted the moon and earth alike 
and in parallel lines., it would not disturb the moon's motion Disturbing 
around the earth in the slightest desfree, however powerful its *°^^® °^ ^^^ 

^ o i- upon moon 

attraction might be. The sun always attracts the moon more only one 

than twice as powerfully as the earth does ; but the sun's ^^^i^etieth 

disturbing force upon the moon when at its very maximum is attraction. 
only one ninetieth of the earth's attraction. 

The tyro is apt to be puzzled by thinking of the earth as fixed while the 
moon revolves around it ; he reasons, therefore, that at the time of new 
moon, when the moon is between the earth and sun, the sun would neces- 
sarily pull her away from us, if its attraction were really double that of the 
earth ; and it would do so if the earth were fixed. We must think of the 
earth and moon as both free to move, like chips floating on water, and of 
the sun as attracting them both with nearly equal power, — the nearer of 
the two a little more strongly, of course. 

324. Second^ it is only by a mathematical fiction that the Disturbed 
"disturbed body" is spoken of as "moving" in an ellipse"; it ^o^^es never 

J L OX move 

never does so exactly. The path of the moon, for instance, strictly in 
never returns into itself. an ellipse. 



fiction of the 
ous ellipse. 

ances such 
only techni- 

tions of 
moon due 
solely to 
the sun. 

But it is a great convenience for the purposes of computation 
to treat the subject as if the orbit were a material wire always 
of truly elliptical form, having the moving body strung upon it 
like a bead, this " orbit " being continually pulled about and 
changed in form and size by the action of the disturbing forces, 
taking the body with it, of course, in all these changes. This 
imaginary orbit at any moment is for that moment a true instan- 
taneous ellipse of determinable form and position, but is con- 
stantly changing. It is in this sense that we speak of the 
eccentricity of the moon's orbit as continually varying and 
its lines of apsides and nodes as revolving. 

The student must be careful, however, not to let this icire theory of 
orbits get so strong a hold, upon the imagination that he begins to think 
of the "orbits" as material things, liable to collision and damage. An 
orbit is simply, of course, the path of a body, like the track of a ship upon 
the ocean. 

325. Thirds the "disturbances" and "perturbations" are such 
only in a technical sense. Elliptical motion is no more natural 
or proper to the moon or to a planet than its actual motion is; 
nor in a philosophical sense is the pure elliptical motion any 
more regular (i.e., " rule-following ") than the so-called " dis- 
turbed " motion. 

We make the remark because we frequently meet the notion that the 
" perturbations " of the heavenly bodies are imperfections and blemishes in 
the system. One good old theologian of our acquaintance used to maintain 
that they were a consequence of the fall of Adam. 

326. Lunar Perturbations. — The sun is the only body which 
sensibly disturbs the moon; the planets are too small and too 
distant to produce directly any effect which can be noticed, 
though indirectly by their effects on the orbit of the earth they 
make themselves slightly felt — at second-hand, so to speak. 

The disturbing force due to the solar attraction can be easily 
computed at any moment by methods indicated in the General 

cp:lestial mechanics 29T 

Astronomy^ but we shall not enter into that subject. This Disturbing- 
force is continually chanffinsc in amount and direction, and the ^^^^® easily 

'J f^ ^ ^ corniDuted. 

student can readily understand that the accurate calculation of Computa- 
the summed-up effects of such a variable force in changing^ the *^°^ °^ ^^^ 

sffsct coni- 

orbit of the moon and her place in the orbit must be extremely plicated, 
difficult. For the most part, however, the disturbances are 
periodic^ running through their phases and repeating themselves 
at regular intervals, so that they can be expressed by trigono- 
metrical series. Over one hundred of these separate "inequali- 
ties," as they are called, are now recognized and taken account 
of in the construction of the Nautical Almanac. 

We mention a few only of the moon's disturbances, — those 
which are largest and most important, two or three of which, 
especially those which affect the time of eclipses, were discov- 
ered before the time of Newton, though not explained. 

327. Effect on the Length of the Month; Revolution of the Lengthening 
Line of Apsides; Regression of the Nodes. — (1) Effect on Length ^^^^® 
of the Month. On the whole, the action of the sun tends to 
lessen the effect of the earth's attraction on the moon by about 
3- J-Q part, z.e., it virtually diminishes (jl in equation (5) (Sec. 321), 
and this increases ^, the period or length of the month, by about 
Y^Q part. The month is nearly an hour longer than it otherwise 
would be at the moon's present distance from the earth. 

(2) Revolution of the Line of Apsides. According to the Advance of 
" age " of the moon at the time when it passes the perigee or ^p^^^^^- 
apogee, the sun shifts the line of apsides for that month, some- 
times forward (eastward) and sometimes backward ; but in the 

long run the forward motion predominates, and the line moves 
eastward and completes a revolution in 8.855 years. 

(3) The Regression of the Nodes. This has already been Regression 
repeatedly mentioned. Speaking generally, the action of the o^^^^®^- 
sun on the whole tends to draw the plane of the moon's orbit 
towards the ecliptic ; but, much as in the case of precession, 

the effect is not felt in any permanent change of the inclination 


ma:n^ual of astronomy 


The yaria- 


of the orbit, but shows itself in a westward shifting of the node, 
which carries it around once in 18.6 years. 

328. Periodic Inequalities. — (4) The JEvection. This is an 
irregularity which at the maximum puts the moon forward or 
backward in its orbit about li°, and has for its period about one 
and one-eighth years, the time required for the sun to complete 
a revolution from the line of apsides of the moon's orbit to the 
same line again. It is the largest of the moon's periodic pertur- 
bations and was discovered by Hipparchus about 150 B.C. It 
was the only perturbation known to the ancients and may affect 
the time of an eclipse by nearly six hours, making it from three 
hours early to three hours late. 

It depends upon an alternate increase and decrease of the eccen- 
tricity of the mooTbS orhit^ which is always a maximum when the 
sun is passing the line of apsides, and a minimum when half- 
way between them. 

(5) The Variation. This is an inequality with a period of 
one synodic month and reaches its maximum of about 40' at the 
octafits, i.e., the points 45° from new and full moon. At the 
octants following the new and full the moon is about 1^20"* 
ahead of time, and at the octants preceding as much behind time. 

This inequality was detected by Tycho Brahe about 1580, 
though there is some reason to suppose that it had been dis- 
covered some five hundred years before by an Arabian astrono- 
mer (Aboul Wefa) and lost sight of. It becomes zero at the full 
and new moon, and therefore does not affect the time of eclipses. 
For this reason it was missed by the Greek astronomers. 

(6) The Annual liquation. This is the one remaining ine- 
quality which affects the moon's place by an amount percep- 
tible to the naked eye. At the maximum it is about 11', with 
a period of one " anomalistic year " (Sec. 182). 

It depends upon the fact that when the earth is nearest the 
sun, in January, the sun's disturbing effect on the moon is 
greater than the average, and the month is lengthened a little 


more than usual ; and vice versa when the sun is most distant, 
in July. For half the year, therefore, from October to April, 
the moon keeps falling behind^ while in the other half of the 
year the month is slightly shortened and the moon gains. 

For more detailed geometrical explanations the student is referred to 
the General Astronomy and to Herschel's Outlines of Astronomy, or to works 
on celestial mechanics for their analytical discussion. 

329. The Secular Acceleration of the Moon's Mean Motion. — Secular 
Among" the multitude of lesser inequalities of the moon's motion ^^^®^®^- 

. .... ation. 

this is of special interest theoretically and is still in some respects 
a "bone of contention" among astronomers. It was discovered 
by Halley about two hundred years ago. From a comparison 
of ancient with modern eclipses, he found that the month is now 
certainly shorter than it was in the days of Ptolemy, and that 
the shortening has been progressive, the moon at present being 
about a degree, or tw^o hours in time, in advance of the position 
it would occupy if it had kept its motion unchanged since the 
Christian era. So far as astronomers could see at the time of 
the discovery, the process would continue indefinitely, — in secula 
seculorum ; hence the name. 

Laplace about 1800 showed that this effect can be traced to 
the change in the eccentricity of the earth's orbit, which is at 
present diminishing (Sec. 164). Since the major axis remains 
unaffected, decrease of eccentricity implies an increase of the its cause 
hreadth (minor axis) of the ellipse, of its area also, and therefore !^ !^. ^"^^'^" 
of the average distance of the earth from the sun during the eccentricity 
year. From this increased distance between earth and sun fol- ^^ ^^^® 

•^ , , _ earth's 

lows a decreased lengthening of the month by the sun's disturbing orbit. 
action (Sec. 327). This practically amounts to the shortening of 
the month, which shortening will continue as long as the eccen- 
tricity of the earth's orbit continues to diminish, — about 24000 
years, when the effect will cease and be reversed. 

The theoretical amount of this acceleration of the moon's 
mean motion is about 6'' in a century, while the actual value, 



and ob- 

by tidal 

The tides 

that they 
are mainly 
due to action 
of the moon. 

according to different estimates depending on comparison of 
modern with the much less accurate ancient observations, is decid- 
edly larger, — 8''. 09, according to StockAvell. The discrepancy 
is now generally ascribed to a slight lengthening of the day, 
diminishing the number of seconds in a month and so making 
the month apparently shorter, as containing a smaller number 
of seconds. Such a lengthening of the day could be accounted 
for by a retardation of the earth's rotation due to the friction of 
the tides (Sec. 345), but the actual difference which ought to be 
ascribed to this action is as yet very uncertain. 

For an excellent non-technical account of the matter, see Newcomb's 
Popular Astro7wmy, p. 292. 

330. The Tides. — Just as the disturbing force of the sun 
modifies the intensity and direction of the earth's attraction on 
the moon, so the disturbing forces due to the attractions of the sun 
and moon act upon the liquid portions of the earth to modify 
the intensity and direction of gravity and generate the tides. 

These consist in a regular rise and fall of the ocean surface, 
generally twice a day, the average interval between correspond- 
ing high waters on successive days at any given place being 
24^51^^. This is precisely the same as the average interval 
between two successive passages of the moon across the meridian, 
and the coincidence, maintained indefinitely, makes it certain 
that there must be some causal connection between the moon 
and the tides; as some one has said, the odd fifty-one minutes 
is " the moon's earmark." 

That the moon is largely responsible for the tides is also shown 
by the fact that when the moon is in jt^eri^ee, z.e., at the nearest 
point to the earth, they are nearly twenty per cent higher than 
when she is in apogee. The highest tides of all happen when 
the new or full moon occurs at the time when the moon is in 
perigee, especially if this perigeal new or full moon occurs about 
the first of January, when the earth is also nearest to the sun. 


331. Definitions. — While the water is rising it is flood-tide ; Definition 
when falling^ it is ebb. It is high water at the moment when "^^^rms 

. . connected 

the water-level is highest, and low water when it is lowest. The with the 
spring-tides are the largest tides of the month, which occur near ^^^^^^ 
the times of new and full moon, while the v.eap tides are the neap tides, 
smallest and occur at half-moon, the relative heights of spring 
and neap tides being about as 7 to 3. 

At the time of the spring-tides the interval between the cor- 
responding tides of successive days is less than the average. Priming and 
being only about 24^38"^ (instead of 24^51"^), and then the tides ^^ss^^g- 
are said to prime. At the neap 
tides the interval is greater than 
the mean, — about 25^6™, — and 
the tide lags. 

The establishment of a port is 
the mean interval between the 
time of high water at that port 

and the next preceding passage Fig. 120. — The Tides 

of the moon across the meridian. 

The "establishment" of New York, for instance, is 8^13™; The estab- 
2.e., on the average, high water occurs 8^13°^ after the moon ^^shment. 
has passed the meridian ; but the actual interval varies fully 
half an hour on each side of this mean value at different 
times of the month, and under varying conditions of the 

332. The Tide-Raising Force. — If we consider the moon The tide- 
alone, it appears that the effect of her attraction upon the earth, ^^^^^^s 
regarded as a liquid globe, is a tendency to distort the sphere 
into a slightly lemon-shaped form, with its long diameter point- 
ing to the moon, raising the level of the water about 2 feet, both 
directly under the moon and on the opposite side of the earth 
(at A and B, Fig. 120), and very slightly depressing it on the 
whole great circle which lies half-way between A and B. D and 
E are two points on this circle of depression. 



Why tide is 
raised on 
side opposite 
the moon. 

The earth 
not fixed 
while at- 
tracted by 


Why gravity 
is dimin- 
ished both 
under moon 
and on 
side of 

Students seldom find any difficulty in seeing that the moon's 
attraction ought to raise the level at A ; but they often do find 
it yery hard to understand why the level should also be raised 
at B. It seems to them that it ought to be more depressed just 
there than anywhere else. The mystery to them is how the 
moon, when directly underfoot, can exert a lifting force such as 
would diminish one's weight. 

The trouble is that the student thinks of the solid part of the 
earth as fixed with reference to the moon, and the water alone 
as free to move. If this were the case, he 
would be entirely right in supposing that at B 
gravity would be increased by the earth's attrac- 
tion instead of diminished; the earth, however, 
is not fixed, but perfectly free to move. 

333. Explanation of the Diminution of Gravity 
at the Point opposite the Moon. — Consider three 
particles (Fig. 121) at B, C, and A, moving with 
equal velocities, A a, Bh, and Co, but under the 
action of the moon, which attracts A more 
powerfully than C and B less so. Then, if the 
particles have no bond of connection, at the end 
of a unit of time they will be at B', C', and A', 
having followed the curved paths indicated. But 
since A is nearest the moon, its path will be the most curved of the 
three, and that of B the least curved. It is obvious, therefore, 
that the distances of both B and A from C will have been increased ; 
and if they were connected to C by an elastic cord, the cord would 
be stretched^ both A and B being relatively pulled away from C 
by practically the same amount. We say relatively^ because C is 
really pulled away from B^ rather than B from (7, — C being 
more attracted by the moon than B is ; but the moon's attraction 
tends to separate the two all the same, and that is the point. 

334. The Amount of the Moon's Tide-Raising Force. — When 
the moon is either in the zenith or nadir the weight of a body 

Fig. 121. —The 
Tide-Raising Force 


at the earth's surface is diminished by about one part in eight Gravity 

and a half millions, or one pound in 4000 tons. 

At a point which has the moon on its horizon it can be shown 
that gravity is increased by just half as much, or about one 
seventeen millionth. 

The computation of the moon's lifting force at A and B (Fig. 120) is 
as follows : The distance of the moon from the earth's center is 60 earth 
radii, so that the distances from A and B are 59 and 61, respectively. The 
moon's mass is about -J^ of the earth's. Taking g for the force of gravity 
at the surface of the earth, we have, therefore, attraction of moon on 

A = , attraction on C = 5 and attraction on B = 


80 X 592 80 X 602 80 x 6P 

From this we find 

(A -C) = —^ , and (C - B) = ^ 

^ ^ 8 424000 ^ ^ 8 835000 

Several attempts have been made within the last twenty 
years to detect this variation of weight by direct experiment, 
but so far unsuccessfully. The variations are too small. 

The moon's attraction also produces everywhere, except at Moon's 
A, jB, D, and U (Fip-. 120), a tangential force which urg^es the tangential 

p ^ force. 

particles along the surface towards the line AB and powerfully 
cooperates in the tide-making. 

335. The Sun's Tide-Producing Force. — The sun acts pre- 
cisely as the moon does, but, being nearly 400 times as far 
away,^ its tidal action, notwithstanding its enormous mass, is Tidal in- 
less than that of the moon in the proportion of 2 to 5 (nearly). 

fluence of 
tlie sun 

At new and full moon the tidal forces of the sun and moon con- about two 
spire, and we then have the spring-tides, while at quadrature ^^^^^ *^^* 

.7 7 1 jji 'Ti-i-i.iOf ^^^ moon. 

tnei/ are opposed, ana we get the neap tides, their relative heights 
being as (5 + 2) to (5 — 2). The priming and lagging of the 
tides (Sec. 331) is also due to the sun's influence. 

336. Condition for Permanent Tides. — If the earth were 
wholly composed of water, and if it kept always the same 

1 It can be proved that the "tide-producing force " of a body varies inversely 
as the cube of its distance, and directly as its mass. 



tides if 
earth's day 
were a 
month long. 

Effect of 



Tide crest 
90° from 
moon if 
earth were 
covered by 
deep water. 

cannot keep 
up with 
moon, since 
water is not 
deep enough 
and conti- 

face towards the moon (as the moon does towards the earth), 
so that every particle on the earth's surface were always sub- 
jected to the same disturbing force from the moon, then, leaving 
out of account the sun's action for the present, a permanent 
tide would be raised upon the earth, as indicated in Fig. 120. 
The difference between the level at A and D would in this case 
be a little less than 2 feet. 

337. Effect of the Earth's Rotation. — Suppose, now, the earth 
to be put in rotation. It is easy to see that the two tidal waves 
A and B would move over the earth's surface, following the 
moon at a certain angle dependent on the inertia of the water 
and tending to move with a westward velocity precisely equal 
to that of the earth's eastward rotation, — about 1000 miles an 
hour at the equator. The sun's action would produce similar 
tides superposed upon the lunar tides, and about two fifths as 
large ; and at different times of the month these two pairs of 
tides would be differently related, as has already been explained, 
sometimes conspiring and sometimes opposed. 

- If the earth were entirely covered with deep water, the tide- 
waves would run around the globe regularly, and if the depth 
of water were not less than 13 miles, the tide crests, as can be 
shown (though we do not undertake it here), would follow the 
moon at an angle of just 90°. It would be high water precisely 
where it might at first be supposed we should get low water, 
the place of high water being shifted 90° by the rotation of the 

If the depth of the water were, as it really is, much less than 
13 miles, the tide-wave in the ocean could not keep up with the 
moon, and this would complicate the result. Moreover, the con- 
tinents of North and South America, with the southern antarc- 
tic continent, make a barrier almost complete from pole to pole, 
leaving only a narrow passage at Cape Horn. Consider also 
the varying depth of the water of the different oceans and the 
irregular contours of the shores, and it is evident that the whole 


combination of circumstances makes it quite impossible to deter- 
mine by theory what the course and character of the tide-waves 
must be. We are obliged to depend upon observations, and 
observations are more or less inadequate, because, with the 
exception of a few islands, our only possible tide stations are 
on the shores of continents where local circumstances largely 
control the phenomena. 

338. Free and Forced Oscillations. — If the water of the 

ocean is suddenly disturbed, as, for instance, by an earthquake, Free waves 
and then left to itself, a " free wave " is formed, which, if the ^^ * ® 


horizontal dimensions of the wave are large as compared with their 
the depth of the water, will travel at a rate depending solely on velocity. 
the depth. 

Its velocity is equal, as can be proved, to the velocity acquired 
hy a body in falling through half the depth of the ocean; i.e., 
V = v^A, where h is the depth of the water. 

Observations upon waves caused by certain earthquakes in South 
America and Japan have thus informed us that between the coasts of 
those countries the Pacific averages between 2|- and 3 miles in depth. 

Now, as the moon in its apparent diurnal motion passes across 
the American continent each day and comes over the Pacific 
Ocean, it starts such a " parent " wave in the Pacific, and a 
second one twelve hours later. These waves, once started, 
move on nearly (but not exactly) like a free earthquake wave, 
— not exactly, because the velocity of the earth's rotation being 
about 1050 miles an hour at the equator, the moon moves 
(relatively) westward faster than the wave can naturally follow 
it, and so for a while the moon slightly accelerates the wave. 
The tidal wave is thus, in its origin, a "forced oscillation" ; in 
its subsequent travel it is very nearly, but not entirely, " free." 

339. Cotidal Lines. — Cotidal lines are lines drawn upon the Cotidai lines 
surface of the ocean connecting points which have their high ® ^^ ' 
water at the same momeyit of Greenwich time. They mark the 



Course of 
the daily 
tide- waves. 

Age of the 
tide when 
it reaches 

crest of the tide- wave for every hour, and if we could map them 
with certainty, we should have all necessary information as to 
the actual motion of the tide-wave. 

Unfortunately we can get no direct knowledge as to the posi- 
tion of these lines in mid ocean ; we can only determine a few 
points here and there on the coasts and on the islands, so that 
much is necessarily left to conjecture. Fig. 122 is a reduced 
copy of a cotidal map, borrowed by permission, with some modi- 
fications, from Guyot's Pliysical Geography. 

340. Course of Travel of the Tidal Wave. — In studying this map we 
find that the main or " parent " wave starts twice a day in the Pacific, off 
Callao, on the coast of South America. This is shown on the chart by a 
sort of oval " eye " in the cotidal lines, just as on a topographical chart the 
summit of a mountain is indicated by an eye in the contour lines. From 
this point the wave travels northwest through the deep water of the Pacific 
at the rate of about 850 miles an hour, reaching Kamchatka in ten hours. 
Through the shallower water to the west and southwest the velocity is only 
from 400 to 600 miles an hour, so that the wave arrives at New Zealand 
about twelve hours old. Passing on by Australia and combining with the 
small wave which the moon raises directly in the Indian Ocean, the result- 
ant tide crest reaches the Cape of Good Hope in about twenty-nine hours 
and enters the Atlantic. 

Here it combines with a smaller tide-wave, twelve hours younger, which 
has backed into the Atlantic around Cape Horn, and it is also modified 
by the direct tide produced by the moon's action upon the Atlantic. The 
tide resulting from the combination of these three then travels northtvard 
through the Atlantic at the rate of nearly 700 miles an hour. It is about 
forti/ hours old, reckoning from the birth of its principal component in 
the Pacific, w^hen it first reaches the coast of the United States in Florida ; 
and our coast is so situated that it arrives at all the principal ports within 
two or three hours of that time. It is forty-one or forty-two hours old 
when it reaches iSI'ew York and Boston. 

To reach London it has to travel around the northern end of Scotland 
and through the N^orth Sea, and is nearly sixty hours old when it arrives 
at that port and at the ports of the German Ocean. 

In the great oceans there are thus three or four tide crests traveling 
simultaneously, following each other nearly in the same track, but with 
continual minor changes. If we take into account the tides in rivers and 





Speed and 
limit of 
ascent of 
tides in 

sounds, the number of simultaneous tide crests must be at least six or 
seven ; i.e., the tidal v^ave at the extremity of its travel (up the Amazon 
River, for instance) must be at least three or four days old, reckoned from 
its birth in the Pacific. 

341. Tides in Rivers. — The tide-wave ascends a river at a 
rate which depends upon the depth of the water, the amount 
of friction, and the swiftness of the stream. It may, and gener- 
ally does, ascend until it comes to a rapid where the velocity of 
the current is greater than that of the wave. In shallow streams, 
however, it dies out earlier. Contrarj^ to what is usually sujj- 
posed, it often ascends to an elevation far above that of the highest 
crest of the tide-wave at the rivers mouth. In the La Plata and 
Amazon it goes up to an elevation of at least 100 feet above 

Height of 
tides in mid 
ocean and 
near shore. 

height of 

H K 

Fig. 123. — Increase in Height of Tide on approaching the Shore 

the sea-level. The velocity of the tide-wave in a river seldom 
exceeds 10 or 20 miles an hour, and is usually much less. 

342. Height of Tides. — In mid ocean the difference between 
high and low water is usually between 2 and 3 feet, as observed 
on isolated islands in deep water ; but on continental shores the 
height is ordinarily much greater. As soon as the tide-wave 
'' touches bottom," so to speak, the velocity is diminished, the 
tide crests are crowded more closely together, and the height 
of the wave is increased somewhat as indicated in Fig. 123. 
Theoretically, it varies inversely as the fourth root of the depth ; 
i.e.^ where the water is 100 feet deep the tide-wave should be 
twice as high as at the depth of 1600 feet. 

Where the configuration of the shore forces the tide into a 
corner it sometimes rises very high. In Minas Basin, near the 
head of the Bay of Fundy, tides of 70 feet are said to be not 
uncommon, and some of nearly 100 feet have been reported. 


343. Effect of the Wind and Changes in Barometric Pressure. — When the Effect of 

wind blows into the mouth of a harbor, it drives in the water by its sur- wind and 

face friction and may raise the level several feet. In such cases the time ^ ^^-^S^s o 

of high water, contrary to what might at first be supposed, is delayed, ^^ height 

sometimes as much as fifteen or twenty minutes. This depends upon the of tide and 

fact that the water runs into the harbor for a longer time than it would do if time of high 

the wind were not blowing. water. 

When the wind blows out of the harbor, of course there is a corre- 
sponding effect in the opposite direction. 

When the barometer at a given port is lower than usual, the level of the 
water is usually higher than it otherwise would be, at the rate of about 
1 foot for every inch of difference between the average and actual heights 
of the barometer. 

344. Tides in Lakes and Inland Seas. — These are small and difficult to Tides in 
detect. Theoretically, the range between high and low water in a land- lakes, 
locked sea should bear about the same ratio to the rise and fall of tide 

in mid ocean that the length of the sea does to the diameter of the 
earth. On the coasts of the Mediterranean the tide averages less than 
18 inches, but it reaches the height of 3 or 4 feet at the head of some of 
the gulfs. In Lake Michigan, at Chicago, a tide of about 1|^ inches has 
been detected, the "establishment" (Sec. 331) of Chicago being about 
thirty minutes. 

345. Effects of the Tides on the Rotation of the Earth. — If Effect of 

the tidal motion consisted merely in the risinP" and fallingc of ^^^^^ ^^^" 

•^ ° ° ^ length of 

the particles of the ocean to the extent of some 2 feet twice day. 
daily, it would involve a very trifling expenditure of energy, 
and this is the case with the mid-ocean tide. But near the land 
this slight oscillatory motion is transformed into the bodily 
traveling of immense masses of water, which flow in upon the 
shallows and then out again to sea with a great amount of fluid 
friction, and this involves the expenditure of a very consider- 
able amount of energy. From what source does this energy 
come ? 

The answer is that it must be derived mainly from the earth's 
energy of rotation, and the necessary effect is to lessen the speed 
of rotation and to lengthen the day. Compared with the earth's 
whole stock of rotational energy, however, the loss by tidal 



ing causes. 

Effect of 
tide to cause 
the moon's 
distance to 
increase and 
to lengthen 
the month. 



friction even in a century is very small and the theoretical 
effect on the length of the day extremely slight. Moreover, 
while it is certain that the tidal friction, h?/ itself considered, 
lengthens the day, it does not follow that the day grows longer. 
There are counteracting causes, — for instance^ the earth's 
radiation of heat into space and the consequent shrinkage of 
her volume. At present we do not know as a fact whether the 
day is really longer or shorter than it was a thousand years 

ago. The change, if real, cannot well be as 
great as y-oVo" ^^ ^ second. 

346. Effect of the Tide on the Moon's 
Motion. — Not only does the tide diminish 
the earth's energy of rotation directly by the 
tidal friction, but theoretically it also com- 
municates a minute portion of that energy to 
the moon. It will be seen that a tidal wave, 
situated as in Fig. 124, would slightly accel- 
erate the moon's motion, the attraction of the 
moon by the tidal protuberance, F, being 
slightly greater than that of the opposite wave 
at F . This difference would tend to draw it 
along in its orbit, thus slightly increasing its 
velocity, and so indirectly increasing the major 
axis of the moon's orbit as well as its period. 
The tendency is, therefore, to make the 
moon recede from the earth and to lengthen the month. 

Upon this interaction between the tides and the motions of 
the earth and moon Prof. George Darwin has founded his 
theory of tidal evolution; viz., that the satellites of a planet, 
having separated from it millions of years ago, have been made 
to recede to their present distances by just such an action. 

An excellent popular statement of this theory will be found in the 
closing chapter of Sir Robert Ball's Story of i'he Heavens, and one more 
complete, but still popular, in Darwin's Time and Tide. 

Fig. 124. — Effect of 
the Tide on the 
Moon's Motion 


Bode's Law — The Apparent Motions of the Planets — The Elements of their Orbits 
— Determination of Periods and Distances — Perturbations, Stability of the 
System — Data referring to the Planets themselves — Determination of Diam- 
eter, Mass, Rotation, Surface Peculiarities, Atmosphere, etc. — Herschel's Illus- 
tration of the Scale of the System 

347. The stars preserve their relative configurations, however stars 
much thev may alter their positions in the sky from hour to sensibly 

. ,. ,, . fixed on 

hour. The Dipper always remams a " dipper m every part of celestial 
the diurnal circuit. sphere; 

But certain of the heavenly bodies, and the most conspicuous ^^^^ 
of them, behave differently. The sun and the moon move 
always steadily eastward through the constellations ; and a few 
others, which look like brilliant stars, but are not stars at all, 
creep back and forth among the star groups in a less simple 

These moving bodies were called by the Greeks Planets^ i.e., 
" wanderers." They enumerated seven, — the Sun and Moon 
and, in addition. Mercury, Venus, Mars, Jupiter, and Saturn. 

348. List of Planets. — At present the sun and moon are not List of the 
reckoned as planets ; but the number of others known to the pi^^^ts. 
ancients has been increased by two new worlds, — Uranus and 
Neptune, of great magnitude, though inconspicuous on account 

of their distance, — besides a host of little asteroids. 

The list of the principal planets in their order of distance 
from the Sun stands thus at present: Mercury, Venus, the 
Earth, Mars, Jupiter, Saturn, Uranus, and Neptune. 

Moreover, between Mars and Jupiter, where there is a wide Asteroids 
gap in which another planet would naturally be looked for, ^^^ ^^^^' 




Planets non 

there have already (January, 1902) been discovered more than 
five hundred little bodies called " asteroids," which probably 
represent a single planet, somehow " spoiled in the making," 
so to speak, or subsequently burst into fragments. 

One of this family, Eros, discovered in 1898, crosses the 
inner boundary mentioned, — the orbit of Mars, — and at times 
comes nearer to the earth than any other heavenly body except 
the moon. 

The planets are non-luminous bodies which shine only by 
reflected sunlight, — globes which, like the earth, revolve 
around the sun in orbits nearly circular, moving all of them 
in the same direction and (with numerous exceptions among 
the asteroids) nearly in the common plane of the ecliptic. 

All but the inner two and the asteroids are attended by 
satellites. Of these the Earth has one ( the moon). Mars two, 
Jupiter five, Saturn eight (and perhaps nine), Uranus four, and 
Neptune one. 

349. Relative Distances of the Planets from the Sun; Bode's 
Bode's Law. Law. — There is a curious approximate relation between the 
distances of the planets from the sun, usually known as Bode's 

It is this : Write a series of 4's. To the second 4 add 3 ; to 
the third add 3 x 2, or 6 ; to the fourth, 4 x 3, or 12 ; and so 
on, doubling the added number each time, as in the following 
scheme : 





































The resulting numbers (divided by 10) are approximately 
No satis- equal to the true mean distances of the plaiT^ts from the sun, 
factory expressed in radii of the earth's orbit (astronomical units) — 

explanation ^ ^ _ _ ^ ' 

of the law excepting Neptune, however ; in his case the law breaks down 
yet reached, u^tterly. For the present, at least, it must therefore be regarded 



as a mere coincidence rather than a real ''law," but it is not 
unlikely that its explanation may ultimately be found when the 
evolution of the solar system comes to be better understood. 

It is known as Bode's Law because first brought prominently into notice 
by him in 1772, though it appears to have been discovered by Titius of 
Wittenberg some years earlier. 

350. Table of Names, Distances, and Periods 






Sid. Period 


Mercury . , . 
Yenus .... 







- 0.013 
+ 0.023 


- 0.077 

88d or 3n^ 
224^.7 or 7^^ 
365id or ly 
687<i orUlOm 



Mean asteroid 




3.V.1 to 8y.9 


Jupiter .... 
Saturn .... 
Uranus .... 
Neptune . . . 

& ¥ 







+ 0.002 

- 0.417 

- 8.746 






The column headed "Bode" gives the distance according to Bode's Law; 
the column headed "Diff.," the difference between the true distance and that 
given by Bode's Law. 

Table of 
the planets ; 
and periods. 

351. Periods. — The sidereal period of a planet is the time of 
its revolution around the sun, from a star to the same star 
again, as seen from the sun. The synodic period is the time 
between two successive conjunctions of the planet with the sun, 
a% seen from the earth. 

The sidereal and synodic periods are connected by the same 
relation as the sidereal and synodic months (Sec. 191), namely, 

— = , in which ^, P, and S are, respectively, the periods 

of the earth and of the planet's synodic period; and the numer- 
ical difference between — and — is to be taken without regard to 

Definition of 
sidereal and 








sign ; i.e., for an inferior planet, — = ; for a superior one, 

Fig. 125. — The Planetary Orbits 

The two last columns of the table of Sec. 350 give the 

approximate periods, both sidereal and synodic, for the different 


Map of the Fig. 125 shows the smaller orbits of the system (including 

the orbit of Jupiter), drawn to scale, the radius of the earth's 


orbit being taken as one centimeter. 




On" this scale the diameter of Saturn's orbit would be 19.08 
centimeters, that of Uranus 38.36 centimeters, and that of Nep- 
tune 60.11 centimeters, or about 2 feet. The nearest fixed star, 
on the same scale, would be about a mile and a quarter away. 

It will be seen that the orbits of Mercury, Mars, Jupiter, and 
several of the asteroids are quite distinctly eccentric. 

352. Explanation of Terms. — Fig. 126 illustrates the mean- Technical 
ing of various terms used in describing the position of a planet 
with respect to the conjunction 

sun. U in the fig- 
ure is the position 
of the earth, the 
inner circle is the 
orbit of an inferior 
planet (Mercury or 
Venus), and the 
outer circle is that 
of a superior planet, 
Mars^ for instance. 

The Elongation \ x^.^.^.,...^....,. y j Elongation, 

of a planet is the 
angle at the earth 
between lines 
drawn from the ob- 
server to the planet 
and to the sun, 

^.e., the apparent angular distance of the planet from the sun; 
for a planet at F it is the angle 8EF. 

For a superior planet the elongation can have any value from 
0° to 180°. For an inferior planet there is a certain maxi- 
mum value, called the greatest elongation^ which must be less 
than 90°. This greatest elongation is the angle between a line 
drawn from the earth to the sun and another line drawn tangent 
to the planet's orbit, — the angle VES in the figure. 

Fig. 126. — Planetary Configurations and Aspects 



tion : supe- 
rior and 



motion of 
planets com- 
plicated by 

Absolute Conjunction occurs when the elongation of the 
planet is zero ; superior conjunction when the planet is beyond 
the sun ; inferior when between earth and sun, — a position, 
of course, impossible for a superior planet. Conjunction iyi 
longitude occurs when the planet's longitude is the same as the 
sun's, and in right ascension -when it has the same right ascen- 
sion as the sun. 

Opposition occurs when the elongation of a planet is 180° 
and the planet rises at sunset. 

Quadrature occurs when the planet has an elongation of 90°. 
An inferior planet cannot be in either opposition or quadrature. 

The astrologers called these positions "aspects" and recognized 
several others, — for instance, " sextile," "trine," " octant," etc. 

353. Apparent Motions of the Planets. — If we imagine our- 
selves looking down upon the orbits perpendicularly from their 
northern side, so as to see them in plan., they would appear as 
shown in Fig. 125, and the planets would travel regularly for- 
ward (contrary to the hands of a watch) with a steady, almost 
uniform, motion. Viewed from the earth, however, we see the 
orbits nearly edgewise, and their apparent motions are compli- 
cated, being made up of their own real motion around the sun, 
combined with a purely apparent motion due to the movement 
of the earth. 

Their apparent motion as seen by us may be considered under 
three different aspects : 

(1) The motion in space relative to the earth. 

(2) The motion on the celestial sphere relative to the constel- 
lations^ i.e.^ change of right ascension and declination or of 
celestial latitude and longitude. 

(3) With reference to their apparent angular distance from 
the sun., i.e.., motion in elongation. 

354. Motion in Space Relative to the Earth. — The funda- 
mental principle of relative motion is that if we look at a body 
at rest while we ourselves are moving, its relative motion., i.e., the 



change in its distance and directiori from us, will be the same as 
if we were at rest and it possessed our motion reversed. If we 
look at a body while we move to the south, it appears to move 
towards the )wrth. If we appi^oach it, the effect is the same as 
if it ivere co77iing toivards us, and so on. 

If the body has a motion of its own, then the total apparent 
or relative motion will be the residtant of its real motion com- 
bined with our reversed motion, 
according to the law of compo- 
sition of motions [Physics, 
p. 18). 

A planet at rest, therefore, 
would appear to move in an 
orbit precisely like that of the 
earth in form and size and in 
the same plane, always keeping 
its motion opposed to our own, 
though going around this ap- 
parent orbit in the same direc- 
tion as the earth (just as any 
two opposite points on the 
circumference of a revolving 

wheel are always moving in opposite directions, though going 
the same way around the axis). And since the planets are 
really revolving around the sun, it follows that their apparent 
or geocentric motion is a combination of two motions, — that of 
a body moving once a year around the circumference of a circle ^ 
equal to the earth's orbit, while at the same time the center of 
that circle is carried around the sun in the real orbit of the 
planet, and in the same period with the planet. Jupiter, for 
instance, appears to move as in Fig. 127. 

This is the orbit we should find if we were to attempt to map 
it out by the method used for determining the form of the orbit 

1 The "circles " spoken of here are strictly ellipses of small eccentricity. 

motion in 
space: a 
of the 
real motion 
with that of 
the earth 

Fig. 127. 

Geocentric Motion of Jupiter 
from 1708 to 1720 
After Cassini 

Result an 
to earth. 



Effect of the 
of the real 

motion on 
the celestial 
sphere : 
direct and 
in right 
and longi- 


of the earth around the sun (Sec. 159), z.e., by observing the 
direction of the planet from the earth, and at the same time 
measuring its apparent diameter in order to get its relative dis- 
tances at different times. Practically, however, the method 
would not succeed very well, since the planet's apparent 
diameter is too small to permit the necessary precision in deter- 
mining the variations of distance. 

A motion of the kind represented in the figure is loosely 
called " epicycloidal," — not quite accurately, because the orbits 
concerned are not true circles, so that the loops are of varying 

The Ptolemaic theory of the solar system was fundamentally 
an acceptance of this apparent motion of the planets relative to 
the earth as real, though his theory involved certain serious 
errors of arrangement and proportion. 

355. Motion of a Planet on the Celestial Sphere, i.e., in Right 
Ascension and Declination, or in Latitude and Longitude. — Look- 
ing at Fig. 127, we see that, viewed from the earth, the planet 
moves most of the time " direct," i.e.., eastward in the direction 
of the arrow, as at the points aa ; but while rounding the loops 
at 55, where it comes nearest the earth, its apparent motion is 
reversed and " retrograde," and at certain points, cc, on each 
side of the loop the planet is " stationary " in the sky, its 
motion at the time being directly towards or from the earth. 

Starting from the time of superior conjunction, when the 
planet is at a, it moves eastward, or " direct," among the stars, 
always increasing its right ascension or longitude, but at a rate 
continually slackening, until at last the planet becomes "station- 
ary" at an eastern elongation from the sun, which depends upon 
the size of the orbit and its distance from the earth. 

From the stationary point it reverses its course and moves 
westward around the loop until it comes to the second stationary 
point at a western elongation, the same (if the real orbit is cir- 
cular) as the eastern elongation of the former stationary point. 



There it resumes its eastward motion and continues it until it 
reaches the next superior conjunction, at the end of a synodic 

The middle of this arc of regression is always very near the 
point where the planet comes nearest the earth, ^.e., at " oppo- pianet 
sition" for a superior planet, and at "inferior conjunction" for retrogrades 
an inferior planet. In time, as well as in the number of degrees nearest 
passed over, the direct motion always exceeds the retrograde in *^® earth, 
each synodic period of the planet. 

As observed with a transit-instrument, all planets when 
moving eastward (direct) come later to the meridian each night 
hy the sidereal clocks and vice versa when retrograding. 

356. Motions in Latitude. — If the orbits of the planets all lay Motion in 

precisely in the same plane with the earth's orbit, their apparent 

latitude ; 

Fig. 128. — Motion of Saturn and Uranus in 1897 

orbits relative to the earth would do so also, and their apparent 
motions on the celestial sphere would be simply forward and 
backward upon the ecliptic. 

But while the orbits of the larger planets are only slightly 
inclined to the ecliptic, so that they never go very far from it, 
they do, in fact, deviate a few degrees one side and the other, 



Motion with 
respect to 
from sun. 

come to 
every day 
by mean- 
time watch. 

SO that their paths in the heavens form more or less complicated 
loops and kinks. Fig. 128 shows the loops made by Saturn 
and Uranus in 1897, when they happened to be very near each 
other in the sky. 

Certain of the " asteroids " have orbits greatly inclined to the ecliptic 
and very eccentric, as, for instance, the little Eros. The description of 
apparent motions as given above would therefore require very serious modi- 
fication in their case. Eros is sometimes found in circumpolar regions 
more than 40° north of the ecliptic ; sometimes its nearest approach to the 
earth does not coincide with the time of its opposition within several 
weeks ; and sometimes at the time 6i its opposition its motion is more 
nearly from north to south than from east to west. 

357. Motion of the Planets in Elongation, i.e., with Respect to 
the Sun's Place in the Sky. — The visibility of a planet depends 
mainly on its elongation, because when near the sun the planet 
will be above the horizon only by day. As regards their motion, 
considered from this point of view, there is a marked difference 
between the inferior planets and the superior. 

(1) The Superior Planets drop always steadily westward with 
respect to the sun\s place in the heavens, continually increasing 
their western elongation or decreasing their eastern. As 
observed by an ordinary timepiece (keeping solar time), they 
therefore invariably rise earlier and come earlier to the meridian 
every successive night, never moving eastward among the stars 
as rapidly as the sun, even when their direct motion is most 
rapid. This relative motion westward with respect to the sun 
is not, however, uniform. It is slowest near superior conjunc- 
tion, most rapid at opposition. 

Beginning at conjunction the planet is then behind the sun, 
at its greatest distance from the earth, and invisible. It soon, 
however, reappears in the morning, rising before the sun as a 
"morning star," and passes on to western quadrature, when it 
rises at midnight. Thence it moves on to opposition, when it 
is nearest and brightest, and rises at sunset. Still dropping 


westward and receding, it by and by reaches eastern quadrature 
and is on the meridian at sunset. Thence it still crawls slug- 
gishly westward as an " evening star," until it is lost in the 
twilight and completes its synodic period by again reaching 

358. (2) The Inferior Planets, on the other hand, apparently inferior 
oscillate across the sun, moving out equal, or nearly equal, ^ ^^^^^ 
distances on each side of it, but making the westward from one 
swing" between us and the sun much more quickly than the side of sun 

° ^ "^ to the other. 


At superior conjunction an inferior planet is moving east- 
ward /asfer than the sun. Accordingly, it creeps out into the 
twilight as an " evening star," and continues to increase its 
apparent distance from the sun until it reaches its greatest east- 
ern elongation (47° for Venus ; for Mercury, from 18° to 28°). 
Then the sun begins to gain upon it, and as the planet itself 
soon begins to retrograde, the elongation diminishes rapidly and The west- 
the planet hurries back to inferior conjunction, passes it, and ^^^^ swing 
then as a "morning star" moves swiftly out to its western swifter 
elongation. There it turns and climbs slowly back to superior *^^^ *^® 


conjunction again. 

359. The Ptolemaic System. — Assuming the fixity and cen- The system 

tral position of the earth and the actual revolution of the ^^P^oiemy. 
■^ ^ ^ The Alma- 

heavens, Ptolemy (who flourished at Alexandria about A.D. 140) gest. 

worked out the system which bears his name. 

In his great work, the Almagest (Arabic for the Greek The 

G-reatest), which for fourteen centuries was the authoritative 

" Scripture of astronomy," he showed that all the apparent 

motions of the planets, so far as then observed, could be 

accounted for by supposing each planet to move around the 

circumference of a circle called the " epicycle," while the center The epicycle, 

of this circle, sometimes called the " fictitious planet," itself ^^^^^^^^^ 

^ planet, and 

moved around the earth on the circumference of another and deferent. 
larger circle, called the " deferent." 



Error of 
in respect to 
orbits of 
and Venus. 

added by the 
omers. The 

System of 

It was as if the real planet was carried on the end of a crank 
arm which turned around the " fictitious planet " as a center in 
such a way as to point towards or from the earth at times 
when the planet is in line with the sun. 

Fig. 129 represents this Ptolemaic system, except that no 
attention is paid to dimensions, the deferents being spaced at 
equal distances. 

It will be noticed that the epicycle radii, which carry at their extremi- 
ties the planets Mars, Jupiter, and Saturn, are always parallel to the line 
which joins the earth and the sun. 

In the case of Venus and Mercury this was not so. Ptolemy supposed 
that for these planets the deferent circles lay between the earth and the sun, 
and that the fictitious planet in both cases revolved in its deferent once a 
year, always keeping exactly between the earth and the sun ; the motion 
in the epicycle in this case was completed in the time of the planet's period. 
He did not recognize that for these two planets there should be only one 
deferent, viz., the orbit of the sun itself, as the ancient Egyptians are said 
to have understood. 

To account for some of the irregularities of the planets' motions it was 
necessary to suppose that both the deferent and epicycle, though circular, 
are eccentric, the earth not being exactly in the center of the deferent, nor 
the " fictitious planet " in the exact center of the epicycle. In after times, 
when the knowledge of the planetary motions had become more accurate, the 
Arabian astronomers added epicycle upon epicycle until the system became 
very complicated. 

King Alphonso of Spain is said to have remarked to the astronomers 
who presented to him the Alphonsine tables of the planetary motions, 
which had been computed under his orders, that "if he had been present 
at the creation he would have given some good advice." 

360. The Copernican System. — Copernicus (1473-1543) 
asserted the diurnal rotation of the earth on its axis, which was 
rejected by Ptolemy, and showed that it would fully account 
for the apparent diurnal revolution of the stars. He also 
showed that nearly all the known motions of the planets could 
be accounted for by supposing them to revolve around the sun, 
with the earth as one of them, in orbits circular^ but slightly 



out of center. His system, as he left it, was nearly that which 
is accepted to-day, and Fig. 125 may be taken as representing 
it. He was, however, obliged to retain a few small epicycles 
to account for certain of the irregularities. 

Up to this time no one dared to doubt the exact circularity 
of celestial orbits. It was considered metaphysically improper 
that heavenly bodies should move in any but perfect curves, Discovery 
and the circle was regarded as the only perfect one. It ^^ *^® 

° -^ ^ elliptical 

form of 
orbits by 

Fig. 129. —The Ptolemaic System 

was left for Kepler, some sixty years later than Copernicus, 
to show that the planetary orbits are elliptical and to bring 
the system substantially into the form in which we know it 

It was nearly a century before the Copernican system, with 
the improvements of Kepler, finally replaced the Ptolemaic. 
In our oldest American universities. Harvard and Yale, the 
Ptolemaic was for a considerable time taught in connection 
with the Copernican. 



System of 
He could 
not detect 
any parallax 
of the stars 
and con- 
cluded that 
the earth 
must be 
at rest. 

The seven 
elements of 
a planet's 

361. Tychonic System. — Tycho Brahe, who came between Copernicus 
and Kepler, found himself unable to accept the Copernican system for two 
reasons. One was that it was unfavorably regarded by the church, and he 
was a good churchman. The other was the really scientific objection that 
if the earth moved around the sun, the fixed stars all ought to appear to 
move in a corresponding manner, each star describing annually an oval in 
the heavens of the same apparent dimensions as the earth's orbit seen from 
the star. Technically speaking, they ought to have an annual parallax. 

His instruments were by far the most accurate that had ever been made, 
and he could detect no such parallax (although it really existed and can 
now be observed) ; hence, he concluded, not illogically, but incorrectly, that 
the earth must be at rest. 

He rejected the Copernican system, placed the earth at the center of the 
universe, according to the then received interpretation of Scripture, made 
the sun revolve around the earth once a year, and then (this was the pecul- 
iarity of his system) made the apparent orbit of the sun the common defer- 
ent for the epicycles of all the planets, making them to revolve around the 

This theory just as fully accounts for all the motions of the planets as 
the Copernican or Ptolemaic, but like the Ptolemaic breaks down abso- 
lutely when it encounters the aberration of light and the annual parallax 
of the stars, now observable with modern instruments, though not with 
Tycho's. The Tychonic system was never generally accepted, and the 
Copernican was soon firmly established by Kepler and Newton. 

362. Elements of a Planet's Orbit. — These are a set of numer- 
ical quantities, seven in number, which describe the orbit with 
precision and furnish the means of finding the planet's place in 
the orbit at any given time, whether past or future, so far as 
that place depends upon the attraction of the sun alone. They 
are as follows : 

(1) The semi-major axis, a. 

(2) The eccentricity, e. 

(3) The inclination to the ecliptic, i. 

(4) The longitude of the ascending node, Q, . 

(5) The longitude of perihelion, ir. 

(6) The period, P, or else the daily motion, /^. 

(7) The epoch, E. 

THE planp:ts in general 


Of these, the first five pertain to the orbit itself, regarded as 
an ellipse lying in space with one focus at the sun, while two 
are necessary to determine the planet's place in the orbit. 

363. The semi-major axis, a (CA in Fig. 130), defines the Size defined 
size of the orbit and is usually expressed in astronomical units. -^ * ® ^®!^^" 

•^ ^ major axis. 

(It will be remembered that the earth's mean distance from the 
sun is the " astronomical unit.") 

The eccentricity, e, defines the orbit's form. It is a mere Form 

Q defined 

numerical quantity, being the fraction -' obtained by dividing by the 

^ eccentricity 

the distance between the sun and the center of the orbit by the 

Fig. 1.30. —The Elements of a Planet's Orbit 

semi-major axis. In some computations it is convenient to use, 
instead of the decimal fraction itself, the angle (/>, which has e 
for its sine, so that e = sin c/). 

The third element, i, the inclination, is the angle between the inclination 
plane of the planet's orbit and that of the earth. In the figure ^^^ ^^^^^' 

XjXkXq of hocIg 

it is the angle KHO, the plane of the ecliptic being lettered deten 
JEKLF and that of the orbit OBBT. 

The fourth element, Q, (the longitude of the ascending node), 
defines what has been called the "aspect" of the orbit plane, 
i.e., the direction in which it faces. The line of nodes is the line 

plane of 
the orbit. 



The period 
and epoch 
furnish the 
means of 
the place of 
planet in 
its orbit. 

JSfW in the figure (the intersection of the two planes of the 
orbit and ecliptic), and the angle °f ^^is the longitude of the 
ascending node. The planet passes from the lower or southern 
side of the plane of the ecliptic to the northern at the point n 
in its orbit. 

The fifth, and last, of the elements which belong strictly to 
the orbit itself is tt, the so-called longitude of the perihelion, 
which defines the direction in which the major axis of the ellipse 
(the line pA) lies on the plane OBBT. Strictly, tt is not a 
longitude, but equals the sum of the two angles Q, and co; ^.e., 
^Y" SN (in the plane of the ecliptic) plus NSp (in the plane of 
the orbit), both reckoned in the direction of the planet's motion. 
NSp, or o), in the figure is about 210° and ^ Sp is about 315°. 

If we regard the orbit as an oval wire hoop suspended in 
space, these five elements completely define its position, form, 
and size. The plane of the orbit is fixed by the two elements 
numbered three and four, the position of the orbit in this plane 
by number five, the form of the orbit by number two, and finally 
its magnitude by number one. 

To determine where the planet will be at any subsequent 
date we need two more elements : 

Sixth, the periodic time. We must have the sidereal period, 
P, or else the mean daily motion, yu, which is simply 360° divided 
by the number of days in P. 

Seventh, and finally, we must have a "starting-point," the 
Epoch, so called; i.e., the longitude of the planet as seen 
from the sun at some given date, usually Jan. 1, 1850 or 1901, 
or else the precise date at which the planet passed the perihelion 
or the node. 

364. If no force acted on the planets except the sun's attrac- 
tion, these elements would never change, but on account of the 
interaction of the planets they do change ; accordingly, it is 
usual to add in tables of the elements columns giving the 
amount by which each element changes in a century. 


It is to be noted also that if Kepler's third law in its uncor- 
rected form were strictly true, as it is not (Sec. 308), we should 
not need both a and P, for if a is expressed in astronomical 
units, P in years would be simply Va'5. 

The method of determining the position of a planet in its orbit, i.e., oi 
computing an ephemeris, belongs to theoretical astronomy and will not be 
treated here. It is sufficient to say that it is possible from the elements 
of the planets to deduce, for any given time, their actual positions in their 
orbits and their distances and directions from the sun and from each 



365. Since the planetary orbits are, for the most part, nearly 
circular and in the plane of the ecliptic, they are described with 
sufficient accuracy for ordinary pui poses by simply giving the 
planet's period and distance from the sun. We proceed to 
show how these two elements may be determined, but note in General 
passing that there is a general method by which all seven of the ^^^^^^ ^^ 

vt£IIiss lor 

elements of a planet's orbit can ordinarily be deduced together computing 
from three accurate observatiofis of the planet's position, sepa- ^^^ ^^^^ ^^^~ 

, , p I i • 11 !• • '1 ments of a 

rated by a tew weeks interval, though m certain special cases pianet from 

?i fourth observation becomes necessary. three com- 

plete obser- 
This general method involves long and complicated calculation, — ^treated vations. 

in works on theoretical astronomy. It was invented in 1801 by Gauss, 

then a young man of twenty-three, in connection with the discovery of 

Ceres, the first of the asteroids, which, after its discovery by Piazzi, was 

lost to observation by passing into conjunction with the sun. 

366. The observations upon which the calculation for the 
elements of a planet's orbit rest are determinations of the 
planet's right ascension and declination, usually made with 
the meridian-circle, but sometimes by the differential method 
(Sees. 116-117) with the equatorial telescope and micrometer, 
or often at present by photography. 


These observations are, of course, made from the earth's sur- 
face^ and before they can be utilized must be corrected for paral- 
lax, so as to give the geocentric place, i.e., the place the planet 
would occupy if seen from the center of the earth. In many 
cases the geocentric right ascension and declination must, for 
convenient use, be further transformed into celestial latitude 
and longitude. (See Sec. 30 and Appendix, Sec. 702.) 
interpoia- 367. Interpolation of Observations. — It often happens that 

tion of ^g want the place of a planet at some particular moment when 

obsel'vatioiis _ 

to furnish a it Cannot be actually observed, as, for instance, when it is below 
planet's ^]^q horizon. If we have a series of observations made about 
moment ^^^^ time. Say for several days before and after, the place at 
when it any moment included within the time covered by the observa- 
actuaiiv tions Can be determined by interjjolatiofi, and with an accuracy 
observed. exceeding that of any single observation of the series. 

The determination can be made graphically/ by simply plot- 
ting the observations on squared paper with a scale of times as 
abscissas, the observed data being plotted as ordinates, and then 
drawing a curve through the points determined by observation, 
as in so many operation^ of the physical laboratory. Whatever 
can be done graphically can, of course, be worked out still more 
accurately by calculation. The principle is of very extensive 
Heliocentric 368. Helioccntrlc Place. — This is the place of a planet as it 
place. would be seen from the sun and is often wanted in calculations. 

When we have once found the node of the planet's orbit and 
the inclination of the orbit, as well as the planet's distance from 
the sun, the heliocentric place and the distance of the planet 
from the earth can be immediately deduced from the geocentric 
by a simple calculation, which, though not difficult, is rather 
tedious and lies outside the scope of this work. (See Watson's 
Theoretical Astronomy., p. 86.) 

369. Determination of the Sidereal Period of a Planet. — First, 
hy Ohservatio7i of its Node Passage. At the moment when a 


planet crosses its node its latitude., botli geocentric and lieliocen- Determina- 
tric, becomes zero, because the planet is then actually in the ^^^^^^/^ 

■^ "^ planet's 

plane of the ecliptic. From a series of observations of its right period by 
ascension and declination made about that time and reduced to o^'^erva- 
latitude and longitude, both the position oi the node and the when it 
time when the planet crossed the node can be deduced. passes the 

The interval between two successive node passages thus 
determined is the planet's period, — exactly, if the node be sta- 
tionary; very approximately in any case, for none of the nodes 
move rapidly. 

The method is not very satisfactory, however, (1) because the 
planetary orbits cross the ecliptic at so small an angle that the 
latitude is almost zero for many hours, so that the precise sec- 
ond is difficult to determine ; (2) then, also, the periods of the 
more distant planets are too long, — Uranus, 84 years ; Neptune, 
164 years, — too long to wait. 

370. Second^ hy the Mean Synodic Period. The sidereal Period 
Beriod may also be determined by finding" the mean synodic ^^©^©^'"^"^ed 

, ^ -^ "^ ... from obser- 

period of a planet from the dates of two oppositions., widely vations of 
separated in time if possible. ^^^^® ^J-_ 

The exact instant of opposition is found from a series of right 
ascensions and declinations observed about the proper date ; by 
comparing the deduced longitudes of the planet with the corre- 
sponding longitudes of the sun we find easily the precise moment 
when the difference was 180°. When the synodic period is 
found the sidereal is at once given by the equations in Sec. 351, 

viz., — = — — — for an inferior planet, and — = — — — for a 
S F E r^ ' SEP 

superior. In the first case, P = ; in the second, P = — • 

^ S-\-E S-E Necessary 

It will not answer for this purpose to deduce the synodic oppositions 
period from two successive oppositions, because, on account of should be 
the eccentricity of the orbits, both of the planet and of the earth, f®P^^^ ^ 

•^ ' i ' by a lonjj- 

the synodic periods are notably variable. The observations must interval. 



method of 
a planet's 
from the 
sun by two 
tions of its 
by an 
of time 
equal to the 

be sufficiently separated in time to give a good determination 
of the meari synodic period. 

In the case of all the older planets we have observations run- 
ning back nearly two thousand years, so that no difficulty arises 
on this score. For the newly discovered planets the method 
would be seldom available. 

371. Geometrical Method of determining a Planet's Distance 
from the Sun in Astronomical Units. — When we know a planet's 
sidereal period it is easy to determine its distance from the sun 
by two observations of the planet's elongation from the sun^ made 
at dates separated by an interval of exactly one of its periods. 

The elongation, it will be remembered, is the difference 
between the longitude of the planet and that of the sun as seen 
from the earth, and is determined for any given date by a series 
of meridian-circle observations of the planet and sun covering 
that date. 

To determine, for instance, the distance of Mars we must 
have two observations of the planet's elongation, MAS and MCS 
(Fig. 131), separated by an interval of 686.95 days; so that at 
the moment of the second observation from the earth at C the 
planet will occupy precisely the same point in its orbit as when 
observed from A nearly two years before. 

In the figure the two angles at A and C are given directly 
by the observations. The angle at the sun, ASC^ is determined 
from tJie eartKs motion during the elapsed time., which is less than 
two sidereal years by 730.53 days minus 686.95 days, i.e.^ by 
43.58 days; this makes the angle ASC very nearly 43°. 

The sides AS and CS are radii vectores of the earth's orbit, 
accurately known in terms of the mean distance of the earth 
from the sun, which is the astronomical unit. 

In the quadrilateral SAMC we have, therefore, the three 
angles J, S, and C given, and the two sides AS and CS\ we 
can, therefore, proceed just as in Sec. 196 in computing the 
line /^Jf, finding both its length as compared with AS, the 



astronomical unit, and also the planet's direction from the sun, 
given by the angle ASM or CSM, both of which come out in the 
course of the calculation. 

The student can follow out for himself the process by which from two 
elongations of Venus, SA V and SB V, observed at an interval of 225 days, 
SV can be determined. A little modification is necessary from the fact 



Fig. 131. — Determination of the Distance of a Planet from the Sun 

that the point Ȥ falls within the triangle formed by the two positions of 
the earth and planet, instead of outside of it, as in the case of Mars. 

372. From a sufficient number of such pairs of observations This method 
distributed around an orbit it is evidently possible to work out ^^^^^ ^^, 

^ ^ ^ ^ Kepler in 

completely its magnitude and form; and it was precisely in proving the 

this way that Kepler, utilizing the rich mine of data contained ^^^^^ °^ 

in iycnos long series oi observations, proved that the orbit oi an ellipse. 
Mars is an ellipse (and later those of the other planets also) and 



The distance 
of an 
planet deter- 
mined by a 
single obser- 
vation of its 

tions : 
and secular. 

deduced their distances from the sun as compared with that of 
the earth. His Harmonic Law was then discovered by simply 
comparing the periods with the distances. Now that we have 
the Harmonic Law, a planet's approximate mean distance can, 
of course, after its period is known, be much more easily found 
by applying that law than by the geometrical method just 

373. Simple Method of finding the Distance of an Inferior 
Planet. — Tn the case of Yenus, which has an orbit almost per- 
fectly circular, we can use the 
method indicated in Fig. 132. 
When the planet is at its great- 
est elongation the angle at V 
is sensibly a right angle, and 
if we then measure the elonga- 
tion SEV^ we have at once SV 
= SEx sin SEV. 

Mercury's orbit is so eccen- 
tric that the method gives only a 
rough approximation, the angle 
at M not being a right angle ; 

but, by taking many observations distributed all around the 

orbit, an accurate result may be obtained. 

374. Planetary Perturbations. — The attractions of the planets 
for each other slightly disturb their otherwise elliptical motion 
around the sun, but their disturbing forces are, with few excep- 
tions, extremely small, and the resulting perturbations are, as a 
rule, much less than in the case of the moon. The exception is in 
the case of some of the asteroids, which at times come near enough 
to the gigantic Jupiter to be displaced by as much as 8° or 10°. 
The interaction between Jupiter and Saturn also produces appar- 
ent displacements of these planets exceeding half a degree. 

The planetary perturbations are divided into two classes : 
(1) the p)eriodic perturbations, which depend on the positions 

Fig. 132. — Distance of an Inferior Planet 
determined by Observations of its 
Greatest Elongation 


of the planets in their orbits and affect their orbital positions 
(these generally run through their cycle within a century) ; 
(2) the secular perturbations, which depend on the relative posi- 
tions of the orbits themselves with reference to each other and 
produce changes m the elements of the orbits affecting the positiotis 
of the planets only indirectly (these have periods of thousands, 
and even millions, of years). 

375. Periodic Perturbations. — Those of Mercury never exceed Amount of 
15", as seen from the sun. Those of Venus may reach 30''; ^^^^ periodic 

, perturba- 

those of the earth about 1' (say 30000 miles), and those of Mars tiousof the 
a little exceed 2'. As already mentioned, the mutual disturb- ^i^ei'^wt 


ances of Jupiter and Saturn are much larger, reaching 28' and 
48', respectively. Those of Uranus never reach 3', as seen from 
the sun, and those of Neptune are smaller yet. 

The great perturbation between Jupiter and Saturn is called a "long 
inequality," having a period of 913 years. It is due to the near commen- 
surability of their periods, seventy-seven of Jupiter's periods being almost 
exactly equal to thirty-one of vSaturn's. Betw^een Uranus and Neptune 
there is a somewhat similar "long inequality" with a period exceeding 
4000 years. 

376. Secular Perturbations. — These, as already said, depend Secular per- 
upon the relative positions of the orbits, but not of the planets 
themselves, and their effects are to change the orbits and only orbits. 
indirectly to alter the positions occupied by the planets. These 

secular perturbations are extremely slow in their development, 
running on, as the name implies, ''from age to age." 

A most remarkable fact, first proved by Laplace and 
Lagrange about a century ago, is that the major axes and Constancy 
periods are never altered by these " secular perturbations." ^^ ^^i^^' 

^ , _ _ "^ ^ ^ ^ axes and 

While subject to slight periodical changes, they remain abso- periods. 
lutely constant in the long run, so far as planetary action goes. 

The nodes and perihelia, on the other hand, move around Revolution 

of the 

of nodes 
■i, ciiiLi Mil Liit; penueiici {^imxi 

of Venus alone excepted) advance. 

continuously ; all the nodes regress, and all the perihelia (that 



of inclina- 
tions and 

Stability of 
system not 
affected by 

According to Leverrier, the shortest of these periods of revo- 
lution is 37000 years (the line of nodes of Uranus), and the 
longest is 540000 (that of the perihelion of Neptune). But these 
numbers must not be accepted too confidently, since the rate of 
motion is not constant, but itself is subject to secular variation. 

The inclinations of the orbits to the ecliptic are all slightly 
changed in an irregular oscillatory manner, some increasing and 
some diminishing. As Laplace and Leverrier h^ve proved, 
all these changes are for the principal planets confined within 
narroiv limits of not more than a degree or two. 

The eccentricities also change in the same irregular way, some 
increasing, some decreasing, but never changing greatly. These 
oscillations, both for inclinations and eccentricities, usually 
occupy from 10000 to 50000 years, but change continually; 
a long and extensive swing in one direction may or may not be 
followed by a short one reversing its effects. 

The statements made with reference to the unimportant character of 
the planetary pertm^bations do not apply in the case of the asteroids, the 
orbits of which may possibly be subject to very material alterations. 

377. Stability of the Planetary System. — It was near the end 
of the last century that Laplace and Lagrange succeeded in 
showing that the orbits of the principal planets of the system 
would never be seiiously changed by their mutual attractions. 
As has been set forth in the preceding section, changes occur, 
but none of such a character or extent as to be subversive. 

Since the major axes and periods remain constant, and the 
revolution of the nodes and apsides is of little moment in 
affecting the conditions of the planets, the eccentricities and 
inclinations alone come into consideration ; and with respect to 
these Laplace proved that the changes due to perturbations 
must be too small to be of any serious consequence. 

It does not follow, however, that because the mutual attrac- 
tions of the planets cannot seriously derange the system it 


is, therefore, of necessity securely stable. There are other other causes 
conceivable actions which might end even in its ultimate "1!^\^.^ 

o ... affect it. 

destruction ; such, for instance, as that of a resisting medium^ 
or the intervention of large bodies coming from without. 

378. The <* Invariable Plane." — There is no reason why the Theinvari- 
ecliptic — the plane of the earth's orbit — should be made the ^^p^^^^ 
fundamental plane of reference for the solar system, except, 
that we terrestrials live on the earth. There is in the system, 
however, an "invariable plane," as discovered by Laplace in 
1784, the position of which remains forever unchanged by any 
mutual action between the bodies of the system, just as does 
their common center of gravity. 

This plane is defined by the following conditions : if from all the 
planets perpendiculars are drawn to it (technically, if the planets be 
" projected " on this plane, which passes through the center of gravity of the 
system), and if then we multiply the mass of each planet by the area which 
its projected radius vector describes upon this plane, in a unit of time, 
around the center of gravity, the sum, of fhe products icill he a maximum. 

The determination of the exact position of this jilane demands, however, Position of 
an accurate knowledge of the masses and motions of all the planets, dis- the invari- 
covered and undiscovered, belonging to the system, and the data now in ^ ® ^ ^"®* 
our possession hardly warrant a final assignment of its location. Accord- 
ing to the computation of Stockwell, it is inclined to the present ecliptic at 
an angle of about 1° 55', its ascending node on the ecliptic being in longi- 
tude 106° 56^ As might be expected, it lies between the planes of the 
orbits of Jupiter and Saturn, and very near to that of Jupiter. 


In discussing the " personal peculiarities " of the planets we 
have to consider a variety of different data, mostly obtained 
by telescopic study and micrometric measurements, — such, for 
instance, as their diameters ; their masses and densities ; their 
axial rotation; their surface markings; their atmospheric phe- 
nomena, if any ; their albedo, or light-reflecting power ; and, 
finally, their satellite systems. 



tion of a 
diameter by 

The planet's 
radius, p. 

Surface area 
equals p'^, 
equals p^. 

effect of 
tional error 
in case of 

mass deter- 
mined by 
means of 

379. Determination of <<Size," — Diameter, Surface, and 
Volume. — The size of a planet is found by measuring its appar- 
ent diameter in seconds of arc with some form of " micrometer " 
(Sec. 71) attached to a powerful telescope. Since from the ele- 
ments of the orbit of a planet and of the earth we can find the 
distance of the planet from the earth at any time in astro- 
nomical units, we can at once deduce the real linear diameter 
from the apparent diameter D" by an equation slightly modi- 
fied from that given in Sec. 10, viz., 

A X D" 
linear diameter = A sin D'\ or ^^^_^^, , 


A being the distance of the planet from the earth. This will 
give the linear diameter as a fraction of the astronomical unit 
and can be converted into miles by simply multiplying it by 
93 000000, the number of miles in the unit. 

For many purposes it is convenient to express the planet's 
radius in terms of the earth's radius by dividing half the diame- 
ter in miles by 3959 (the number of miles in the mean radius 
of the earth), designating this relative radius by p. 

The surface area of the planet in terms of the earth's surface 
is then p^, and the volume or hulk of the planet is p^ in terms of 
the earth's volume ; i.e.^ if, as is nearly true in the case of 
Jupiter, p = 11, then the surface of the planet is 121 times that 
of the earth, and its bulk 1331 times that of the earth. 

The nearer the planet, other things being equal, the more 
accurately p and the quantities derived from it can be deter- 
mined. An error of O'M in measuring the apparent diameter 
of Venus when nearest counts for less than thirteen miles, but 
in the case of Neptune it would correspond to more than 1300. 

380. Mass, Density, and Surface Gravity. — If the planet has 
a satellite, its mass compared with the sun is very easily and 
accurately found from the proportion 

S ^ p : p + s 


in which >S' is the mass of the sun, p that of the planet, A the 
mean distance of the planet from the sun, and T the planet's 
sidereal period ; while s is the mass of the satellite, a its mean 
distance from the planet, and t its sidereal period. 

In almost all cases p in the first term may be neglected as 
compared with /S', and in the second term s as compared with p^ 
which makes the proportion read 

43 ^3 ^3 ^2 

^''P'-''Y^'j2'^ whence, p = Sxj^x—. 

If we want the mass as compared ivith the earth, the first Mass of 
proportion becomes . planet com- 

^ ^ pared with 

(earth -(- moon) : (planet + satellite) the earth. 

/ cube of moon's distance \ 
\square of moon's sidereal periody 

/cube of satellite's distance \ 
\square of satellite's period/ 

The mass of the moon being J^ of that of the earth, it cannot 
be neglected in comparison with the earth's mass. (No other 
satellite has a mass more than -^-^-q-q of its planet.) 

It is to be noted also that instead of the actual sidereal period 
of the moon we must use a period about an hour shorter, in 
order to allow for the action of the sun (Sec. 327, (1)). 

The observations upon which this method of determining a Data for 
planet's mass depend are those of the satellite's greatest elonqa- ^^^^^ ^^^^ 

. J, . . . . . satellite's 

tion, the measures of distance being especially important, since distance and 

the distance enters into the formula by its cube. period. 

When a planet has no satellite, as is the case with Mercury Mass of 

and Venus, its mass can be determined only by means of the ^^^^^^ 

•^ "^ determined 

perturbations it produces in the motions of other planets, or of by the per- 

comets that happen to come near it. turbations 

XI p -MT • '^^ • it causes. 

In the case of Mercury the mass is still very uncertain. 

Venus, however, disturbs the earth sufficiently to give a very 

good determination of her mass. 



tion of the 
formula for 

mass -^ p^. 




331. The proportions given in the preceding section are easily 
derived for circular orbits from the equation for the general 
equation of the motion of a small body revolving around a larger, 


{M-\- m) 


G t^ 


This equation is obtained by combining the equation for the 
gravitational attraction between two spheres expressed as an 
acceleration (Sec. 146), viz., m-\- 


with the expression for the central force in circular motion 
(Sec. 306), viz., ^ ^2^ 

/ = 


Replacing D in the first equation by r, and equating the two 
values of /, we have ^.^ , 


= 4 



from Avhich equation (1) follows at once. 

In forming the proportion the constant factor drops out, and 

we have 

M^ + m^ : M^ + m^ 

f 2 

t 2 

As Newton proved, this is accurately true for elliptical orbits 
also if for r we put a, the semi-major axis of the orbit ; but 
the demonstration lies beyond our scope. 

382. Surface Gravity and Density. — When the mass has been 

determined the surface gravity and density follow at once. 

Putting 7 for surface gravity as compared with the earth, we 




m being the planet's mass, and p its radius as compared with 
the earth's. The density, compared with the earth, is simply 


— ; if we want the specific gravity, i.e., density as compared with 


water, we must multiply the result by 5.53, the density of the 
earth. Any error in the measured diameter of course affects 
very seriously the computed density and gravity. 

383. Rotation Period and Data connected with it. — The length Rotation 
of the planet's ''day," when it can be determined at all, is usually p^"°.*^ ^^^^ 

^ . . . position or 

ascertained by observing some well-marked spot on its disk and planet's 
noting^ the times of its successive returns. An approximate ^qnator 

^ .... .J. determined 

value of the rotation period is obtained from the observation of ^y observa- 
such returns during a few days or weeks, and this is afterwards ^^^^^ ^^ ^P"^'^ 
corrected by data furnished from observations separated by the surface. 
longest interval obtainable, — a century or more if possible. 

Mars, however, is the only planet of which the rotation period 
is known with great accuracy ; the others either show no well- 
defined markings, or only such markings as seem to be more or 
less movable on the planet's surface, like spots on the sun. 

In reducing the observations account has to be taken of the 
continual change in the direction of the planet from the earth 
and also of the variations of its distance, which alter the time 
taken by light to reach us. 

In the case of the little planet Eros, a large and regular vari- Rotation 
ation in its brightness, observed for some months earlv in 1901 p®^^*^^ ^^ 

^ . . '^ Eros deter- 

in a certain portion of its orbit, was probably due to its axial mined from 

rotation ; if so, the photometrically measured period of variation sanations 

of brightness gives a determination of the length of its day. brightness. 

(See Sec. 428.) The planet is far too small to show a disk in 

the telescope, and of course no observations of spots are possible. 

The inclination of the planet's equator to the plane of its 
orbit and the positions of its poles and equinoxes are deduced 
from the observations of the paths of the spots as they cross the 
disk. Such data, however, are available only in the cases of 
Mars, Jupiter, and Saturn. 

It may be added that the disappearance of the variations of 
the brightness of Eros in May, 1901, after persisting over two 
months, is naturally explained by Professor Pickering as due to 



from obser- 
vations of 
diameters ; 
also from 
tions of its 

by photo- 


and topog- 

the fact that in May its pole was turned towards us; and, if so, 
this gives us the position of the planet's axis and equator. 

The oblateness, or polar compression, of the planet, due to its 
rotation, is found simply by measuring the difference between 
the polar and equatorial diameters ; but the difference is always 
very small, so that the percentage of its probable error is rather 

In some cases also the oblateness can be determined from 
observation of the motion of the nodes of the planet's satellites. 

384. Data relating to the Light of a Planet. — The bright- 
ness of the planet and the reflecting poAver of its surface, or 
albedo, are determined by observations with the photometer, 
which is sometimes used direct, and sometimes attached to a 
telescope ; we have just pointed out how, in one case at least, 
such observations may also be available for determining the 
rotation of a planet. 

The spectroscopic peculiarities of the planet's light are of course 
studied with a spectroscope, and usually by spectroscopic photog- 
raph}^ A planet always shows, so far as its brightness permits, 
the lines of the solar spectrum and, in some cases, additional 
lines or bands of its own, which give information as to the 
constitution of its atmosphere. 

385. The Planet's Surface Markings and Topography. — These 
are studied with the telescope by making careful notes and 
drawings of the appearances and markings seen at different 
times. If the planet has any well-defined and characteristic 
features by which its rotation can be determined, it is soon 
possible to identify such as are permanent and to chart them 
more or less perfectly. 

At present, however, Mars is the only planet of which we have been 
able to obtain what may be called a real map, though some preliminary 
chartings have been attempted for Venus and Mercury. The surface 
markings, which are often very distinct and beautiful upon Jupiter, are all 
of a more or less transient character. 


Thus far pliotography has given but little help in the study of planetary 
surfaces. The images formed even by the largest telescope are too small 
compared with the " grain " of the sensitive fihn ; and the light of the 
planet is so feeble that long exposure is required, during whicli the atmos- 
pheric disturbances usually confuse the image. It is perhaps not impos- 
sible that in the future these difficulties may be overcome by the invention 
of still more sensitive plates with a finer grain and by the use of telescopes 
of extremely long focal length placed in a fixed position and " fed " by 
a siderostat, — a method suggested, and now being tried, by Professor 
Pickering of Harvard. 

386. The Satellite Systems. — The principal data to be deter- Satellite 
mined in respect to these systems are the distances and periods systems: 

rm ' 1 • • r» 1 the elements 

01 the satellites. These are got by micrometric measures oi the determined 
apparent distance and direction of each satellite from the planet; ^y micro- 
or from other satellites, as is now quite the usual method, since observations. 
the distance and direction between two satellites (which are 
mere points of light) can be measured much more precisely than 
between a satellite and the center of the large disk of a planet. 
The reduction of the observations in this latter case is, however, 
very complicated. 

In a few cases the satellites present disks large enough to be 
measured and show spots upon them, so that questions of their 
rotation and surface markings admit of discussion. Also, where Diameters, 
there are a number of satellites attending a planet, their mutual 
perturbations furnish a very interesting subject of study and 
make it possible to determine their masses relative to that of Masses. 
the planet. 

With the exception of our moon and lapStus (the outer satel- Near 
lite of Saturn), all the satellites move very nearly in the plane 

. ^ J L move nearly 

of their planet's equator, — so far at least as known, since the in plane of 
position of the equators of Uranus and Neptune has never P^^^^^t's 
yet been ascertained. Moreover, all the satellites except the haveorwts 
moon and Hyperion, the seventh .satellite of Saturn, move nearly 
in orbits of very small eccentricity, in fact almost perfect 
circles. Laplace and Tisserand have shown that the " equatorial 




move nearly 
in plane of 

tion of the 

accuracy of 

protuberance" of a planet, due to its axial rotation, compels a 
near satellite to move nearly in the equatorial plane. The more 
distant satellites, like the moon and lapetus, on the other 
hand, move nearly in the orbital plane of the planet. 

The circularity of the satellite orbits is not yet accounted 

387. Classification of Planets. — Humboldt has classified the 
planets in two groups, — the " terrestrial planets," as he calls 
them, and the '^ major planets." The terrestrial group contains 
the four planets nearest the sun, — Mercury, Venus, the Earth, 
and Mars. They are all bodies of similar magnitude, ranging 
from 3000 to 8000 miles in diameter; not very different in 
density and probably roughly alike in physical constitution, 
though probably also differing very much in the extent, density, 
and character of their atmospheres. 

The four major planets — Jupiter, Saturn, Uranus, and Nep- 
tune — are mucb larger bodies, ranging from 32000 to 90000 
miles in diameter ; are much less dense ; and, so far as we can 
make out, present only cloud-covered surfaces to our inspection. 
There are strong reasons for supposing that they are at a high 
temperature, and that Jupiter especially is a sort of " semi-sun " ; 
but this is not certain. 

As to the asteroids^ the probability is that they represent a 
fifth planet of the terrestrial group, Vhich, as has been already 
intimated, failed somehow in its evolution, or else has been 
broken to pieces. 

Fig. 133 gives an idea of the relative sizes of the planets. 
The sun on the scale of the figure would be about a foot in 

388. Tables of Planetary Data. — In the Appendix we present 
tables of the different numerical data of the solar system, derived 
from the best authorities and calculated for a solar parallax of 
8''.80, the sun's mean distance being therefore taken as 92 897000 
miles. These tabulated numbers, however, differ widely in 



accuracy. The periods of the planets and their distances in 
astronomical units are very precisely known ; probably the last 
decimal place in the table may be trusted. Next in certainty 
come the masses of such planets as have satellites, expressed in 
terms of the sun's mass. The masses of Venus and, especially, 
of Mercury are much more uncertain. The distances of the 
planets in miles^ their masses in terms of the earth's mass, and 

Fig. loo. — Relative Sizes of tlie Planets 

their diameters m miles, all involve the solar parallax and 
are affected by the slight uncertainty in its amount. For the 
remoter planets, moreover, diameters, volumes, and densities are 
subject to a very considerable percentage of error, as explained 
above (Sec. 379). The student need not be surprised, therefore, 
at finding serious discrepancies between the values given in 
these tables and those given by other authorities, amounting in 
some cases to ten per cent or twenty per cent, or even more. 
Such differences merely indicate the actual uncertainties of our 



Sir John 
of the scale 
of the solar 

389. Sir John HerschePs Illustration of the Dimensions of the 
Solar System. — In his Outlines of Astronomy Herschel gives 
the following illustration of the relative magnitudes and dis- 
tances of the members of our system : 

Choose any well-levelled field. On it place a globe two feet in diameter. 
This will represent the sun. Mercury will be represented by a grain of 
mustard seed on the circumference of a circle 164 feet in diameter for its 
orbit; Venus, a pea, on a circle of 284 feet in diameter ; the Earth, also 
a pea, on a circle of 430 ; Mars, a rather large pin's head, on a circle of 
654 feet; the asteroids, grains of sand, on orbits having a diameter of 1000 
to 1200 feet ; Jupiter, a moderate-sized orange, on a circle nearly half a mile 
across ; Saturn, a small orange, on a circle of four-fifths of a mile ; Uranus, 
a full-sized cherry or small plum, upon a circumference of a circle more 
than a mile in diameter ; and, finally, Neptune, a good-sized plum, on a 
circle about 2^ miles in diameter. 

We may add that on this scale the nearest star would be on 
the opposite side of the earth, 8000 miles away. 


1. What is the mean daily gain of the earth on Mars as seen from the 
sun, i.e., the synodic motion of Mars, assuming their sidereal periods as 
365.25 days for the earth, and 687 days for Mars? 

2. Find the synodic period of Venus, her sidereal period being 225 days. 

3. Given the synodic period of a planet as 3 years, what is its sidereal 
period? ^^^^ I I of a year, or 

(. 1|- years. 

4. Given a synodic period of 4 years, find the sidereal period. 

5. What would be the sidereal period of a planet which had its synodic 
period equal to the sidereal? Ans. 2 years. 

6. Within what limits of distance from the sun must lie all planets having 

synodic periods longer than 2 years? (Apply Kepler's third law after 

finding the sidereal periods that would give a synodic period of 2 years.) 

A ( 0.763 astron. units, or 70 950000 miles, and 

Ans. ^ ' 

.588 astron. units, or 147 500000 miles. 




7. A brilliant starlike object was seen about 7 p.m. on May 1 exactly at 
the east point of the horizon. Could it liave been one of the planets ? 
If not, why not ? 

8. Mercury was at inferior conjunction on Eeb. 8, 1896, at 1 p.m. 
On May 6, at 15 minutes after noon (exactly one sidereal period later), 
its elongation from the sun was observed to be 18° 50' E. Find the dis- 
tance of the planet from the sun in astronomical units, the earth's orbit 
being regarded as circular. (See Sec. 371.) 

(The fact that the first observation was made at conjunction greatly 
simplifies the calculation.) 



Distance from the sun, 0.835 astron. units. 
(The planet was near perihelion.) 

9. At a time when Jupiter's distance from the earth was 4.6 astronomi- 
cal units its apparent equatorial diameter was observed to be 43". 3. Find 
the diameter in miles as determined by this observation. 

Ans. 89700 miles. 

New Physical Observatory, Greenwich 



Mercury, Venus, and Mars — The Asteroids — Intramercurial Planets- 

Zodiacal Light 

Two names 
lor Mercury. 

Best times 
for seeing 

of Mercury's 
orbit. Excep- 
tional in many 


390. Mercury has been known from remote antiquity, and 
we have recorded observations running back to 264 B.C. At 
first astronomers failed to recognize it as the same body on the 
eastern and western side of the sun, and- among the Greeks it 
had for a time two names, — Apollo when morning star, and 
Mercury when evening star. It is so near the sun that it is 
comparatively seldom seen with the naked eye (Copernicus is 
said never to have seen it), but when near its greatest elonga- 
tion it is easily enough visible as a brilliant star of the first mag- 
nitude, though always low down in the twilight. It is best 
seen in the evening at such eastern elongations as occur in March 
and April. As a morning star it is best seen at western elonga- 
tions in September and October. 

It is an exceptional planet in various ways. It is the nearest 
to the sun, receives the most light and heat^ is the swiftest in its 
movement^ and (excepting some of the asteroids) has the most 
eccentric orbit^ with the greatest inclination to the ecliptic. It is 
also the smallest in diameter (again excepting the asteroids) and 
has the least mass. 

391. Its Orbit. — Its mean distance from the sun is about 
36 000000 miles, but the eccentricity of its orbit is so great 
(0.205) that the sun is 7 500000 miles out of the center, and 
the distance of the planet from the sun ranges all the way from 



28 500000 to 43 500000, while the velocity in its orbit varies 
from 36 miles a second at perihelion to only 23 at aphelion. 
Its distance from the earth ranges from about 50 000000 miles 
at the most favorable inferior conjunction to about 136 000000 
at the remotest superior conjunction. 

A given area upon its surface receives on the average nearly 
seven times as much light and heat as the same area on the 
earth ; and the heat received at perihelion is greater than that 
at aphelion in the ratio of 9:4. For this reason, even if the 
planet's equator should be found to be parallel to the plane of 
its orbit, there must be two seasons in each Mercurian year. Seasons due 
due to the changcing^ distance; and if the planet's equator is *^*^® 

^ ^ eccentricity 

inclined nearly at the same angle as ours, the seasons must be of its orbit 
extremely complicated. 

The side7'eal period is 88 days, and the synodic period (from 
conjunction to conjunction) 116 days. The greatest elongation 
ranges from 18° to 28°, on account of the eccentricity of its 
orbit, and occurs about 22 days before and after inferior conjunc- 
tion. The inclination of the orbit to the ecliptic is about 7°. 

392. The Planet^s Magnitude, Mass, etc. — The apparent Diameter, 
diameter of Mercury rang^es from 5'' to about 13'', according to "^^^^ ^^^■' 

^ ^ ^ _ ' ° of Mercury. 

its distance from us, and the real diameter is very nearly 3000 
miles. Its surface is about one seventh that of the earth, and its 
volume^ or hulk^ one eighteenth. 

The planet's mass is not accurately known ; it is very difficult 
to determine, since it has no satellite, and it is so near the sun 
that its disturbing effect upon the other planets is extremely 
small, so that the values calculated from perturbations produced 
by it are very discordant. Different computers give results rang- 
ing all the way from ^ of the earth's mass to -^-^. It probably lies 
somewhere between ^-^ and J^- I^^ mass is, however, unques- 
tionably smaller than that of any other planet, asteroids excepted. 

Our uncertainty as to its mass prevents us from assigning Small sur- 
any certain values to its density or surface gravity ; probably it ^^^e gravity. 



and phases. 

is not quite so dense as the earth. Assuming Newcomb's mass 
of 2^Y t^^^^ ^^ ^^^ earth, the density comes out about 0.85, and 
its surface gravity a little less than i. 

393. Telescopic Appearances, Phases, etc. — In the telescope 
the planet looks like a little moon, showing phases precisely 
similar to those of the moon. At inferior conjunction the dark 
side is towards us, at superior conjunction the illuminated sur- 
face. At greatest elongation the planet appears as a half-moon. 

of Mercury. 

Fig. 134. — Phases of Mercury and Venus 

It is gibbous between superior conjunction and greatest elonga- 
tion, while between inferior conjunction and elongation it shows 
the crescent phase. 

Fig. 134 illustrates the phases of Mercury (and of Venus also). 

The atmosphere of the planet cannot be as dense as that of 
Venus, because at a transit across the sun it shows no encircling 
ring of light, as Venus does (Sec. 401). Both Huggins and 
Vogel, however, report spectroscopic observations which imply 
the presence of water vapor; z.e., the planet's spectrum, in addi- 
tion to the ordinary dark lines belonging to the spectrum of 
reflected sunlight, shows other bands known to be due to water 
vapor, but it is not yet quite certain whether the vapor is in 
the planet's atmosphere or in our own. On the whole, it is 
probable that the atmospheric conditions are much like those 
upon the moon, since under the powerful action of the solar 



heat a planet of so small a mass would probably lose most of its 
atmosphere, if it ever possessed any. 

Generally the planet is so near the sun that it can be observed 
only by daj^, but when proper precautions are taken to screen 
the object-glass from direct sunlight, its observation is not spe- 
cially difficult. The surface presents very little of interest to 
an ordinary telescope. Like the moon, it is brighter at the edge 
than at the center, 
but until recently no 
markings have been 
observed upon its 
disk well enough de- 
fined to give us any 
trustworthy infor- 
mation as to its 
geography or even 
its rotation. 

394. The Planet's 
Rotation . — Schr oter, 
a German astrono- 
mer and a contempo- 
rary of Sir William 
Herschel, and, to 
speak mildly, an im- 
aginative man, early 
in the last century reported certain observations which he con- 
sidered to indicate high mountains on the planet and deduced 
a rotation period of 24^5™, — a result that stood uncontradicted 
until about 1890 and still appears in many text-books, though 
unconfirmed by other observers with instruments certainly much 
better than his. 

In 1889 the Italian astronomer, Schiaparelli, announced the 
discovery upon the planet of certain dark permanent markings, of 
which he presented a map (Fig. 135). He found also that these 

Fig. 135. — Mercury- 
After Schiaparelli 

Planet best 
with the 
in daytime. 

No satis- 
factory map. 

The planet 
in its 
keeps the 
same face 
the sun. 



Large libra- 
tion in 

markings did not change their positions upon the planet's disk 
even in the course of several hours (a fact obviously inconsistent 
Avith rotation in twenty-four hours), but remained always 
nearly fixed in their position with respect to the " termi- 
nator," — the boundary between the illuminated and unillumi- 
nated hemispheres of the planets. Granting this permanency, 
it follows that the planet rotates on its axis only once during 
its orbital period of eighty-eight days; i.e., it keeps the same 
face always towards the sun as the moon does towards the 
earth. Slight changes in the positions of the spots show, how- 
ever, a comparatively large lihration in longitude (Sec. 203, (2)), 
as there ought to be, considering the great eccentricity of the 
planet's orbit. This libration amounts to about 231-° ; ^.e., the 
sun, seen from a favorable position on the planet, instead of 
rising and setting as with us, must seem to oscillate east and 
west in the sky to the extent of 47° in a period of eighty- 
eight days. 

Schiaparelli's reported discovery excited great interest, but the observa- 
tions are extremely difficult even under the Italian atmosphere, and confir- 
mation was tardy. In 1896, however, Mr. Lowell reported its complete 
corroboration as the result of observations at his Flagstaff Observatory, 
though it is rather difficult to reconcile his drawings of the surface mark- 
ings with those of Schiaparelli. Partial confirmations have also been 
received from other quarters. 

If this rotation period is correct, as it probably is, one face of 
the planet is always sunless and probably intensely cold, while 
the opposite is always exposed to a sevenfold African blaze of 
sunbeams. Between these regions is a space in which, as a 
consequence of librations, the sun alternately rises above the 
horizon and drops back again. 
Albedo ex- 395. Albcdo. — The reflecting power of the planet's surface 
tremeiyiow. ^g ^^^^ j^q^^ — according to Zollner, 0.13, a little less than that 
of the moon and much below that of any other planet, hardly 
higher than that of a darkish granite. 



In the proportion of light given out at its different phases it 
behaves like the moon, flashing out strongly near the " full," 
z.e., near superior conjunction, — a fact which probably indicates 
a rough surface with very little atmospheric absorption of 

396. Transits of Mercury. — At the time of inferior conjunc- Transits of 
tion the planet usually passes north or south of the sun, the ^ercmym 

. . . . . . . May and 

inclination of its orbit being 7° ; but if the conjunction occurs November. 
when the planet is very near its node^ it crosses the disk of the May transits 
sun as a small black spot, — not, however, large enough to be 
seen without a telescope. Since the earth passes the planet's 
node on May 7 and November 9, transits can occur only near 
those dates. 

If the planet's orbit were truly circular, the transit limit 
(corresponding to the ecliptic limit. Sees. 286 and 293) would 
be 2° 10', and the conditions of transit would be the same at 
each node ; but at the May transits the planet is near its aphe- 
lion and exceptionally near the earth, so that the May transits 
are only about half as numerous as the other. 

For the November transits the interval is sometimes only intervals 
7 years, but is usually 13 or 46 years. For the May transits the 
7-year interval is impossible. Twenty-two synodic periods of 
Mercury are pretty nearly equal to 7 years ; 41 much more 
nearly equal to 13 years, and 145 are almost exactly equal to 
46 years. Hence, 46 years after a given transit another one at 
the same node is almost certain. 

The last transit was in November, 1894, and was completely visible in 
the United States. During the first half of the present century transits 
will occur as follows : 

Nov. 12, 1907, May 7, 1924, May 10, 1937, 

Nov. 6, 1914, Nov. 8, 1927, Nov. 12, 1940. 

Only the two first of these will be visible in the United States, and not 
the entire transit in either case. The first transits of which the whole will 
be visible here occur on Nov. 13, 1953, and Nov. 6, 1960. 



ma:n^ual of astroxomy 

Transits of 
Mercury a 
test of uni- 
formity of 

Transits of Mercury are of no special astronomical impor- 
tance, except as furnishing accurate determinations of the 
planet's place. 

Newcomb has made a thorough examination of all the 
recorded transits in order to test the uniformity of the earth's 
rotation. They appear to indicate certain small irregularities 
in it, but hardly establish the fact as absolutely certain. 

Brilliance of 

of orbit of 
Venus. Its 
smaller than 
that of any 
other j)lanet. 


The next planet in order from the sun is Venus, by far the 
brightest and most conspicuous of all, — the earth's twin sister 
in magnitude, density, and general constitution, if not in other 
physical conditions. Like Mercury, it had two names among 
the Greeks, — Phosphorus as morning star, and Hesperus as 
evening star. 

It is so brilliant that it is easily seen by the naked eye in the 
daytime for several weeks when near its greatest elongation; 
occasionally it is bright enough to catch the eye at once, but 
usually is seen by daylight only when one knows precisely 
where to look for it. 

397. Distance, Period, and Inclination of Orbit. — Its mean 
distance from the sun is 67 200000 miles. 

The eccentricity of the orbit is the smallest in the planetary 
system (only 0.007), so that the whole variation of its distance 
from the sun is less than a million miles. 

Its orhital velocity is 22 miles per second. 

The heat and light received from the sun are most exactly 
double the amount received by the earth. 

Its sidereal period is 225 days, or nearly seven and one-half 
months, and its synodic period 584 days, — a year and seven 
months. From superior conjunction to elongation on either 
side is 220 days, while from inferior to elongation it is only 
72 days, — less than one third as long. 


The greatest elongation is 47° or 48°. 
The inclination of its orbit is about S^°. 

398. Magnitude, Mass, Density, etc The apparent diameter Diameter, 

of the planet ranges from 61" at the time of inferior conjunction "^^^^/ 

to only 11" at superior conjunction, the great difference depend- surface 
ing upon the enormous variation in the distance of the planet g^'^^^^y* ®*^- 
from the earth, which is only 26 000000 miles at inferior conjunc- 
tion and 160 000000 at superior. The real diameter of the planet 
is about 7600 miles, according to the recent measures of See at 

According to this, its surface, compared with that of the 
earth, is 0.91 ; its volume, 0.87. (These numbers differ some- 
what from those given in the tables in the Appendix, which are 
allowed to stand unchanged, as illustrating the discrepancies 
between good authorities in such cases.) 

By means of the perturbations she produces upon the earth, 
the mass of Venus is found to be a little more than four fifths 
(0.82) of the earth's ; hence, her density is about ninety-four 
per cent and her superficial gravity ninety per cent of the 
earth's. A man who weighs 160 pounds here would weigh 
about 140 pounds on Venus. 

399. Phases. — The telescopic appearance of the planet is Phases of 
striking on account of her great brilliance. When midway ^®^^^- 
between greatest elongation and inferior conjunction she has an 
apparent diameter of 40'', so that, with a magnifying power of 

only 45, she looks exactly like the moon four days old, and of 
precisely the same apparent size, though very few persons would 
think so on first viewing the planet through a telescope. The 
novice always underrates the apparent size of a telescopic 
object, because he instinctively adjusts his focus as if looking 
at a picture or a page only a few inches away, instead of pro- 
jecting the object visually into the sky. 

According to the theory of Ptolemy, Venus could never 
show us more than half her illuminated surface, since, according 



Phases irrec- to his hypothesis, she was always between us and the supposed 
onciiabie ^^j^^ of the sun. According-lv, when in 1610 Galileo discovered 

with Ptole- "^ ^ ° "^ 

with his newly invented telescope that she exhibited the 
gibbous phase as well as the crescent, it was a strong argu- 
ment for the Copernican theory. 

Galileo announced his discovery in a curious way, by publishing the 

anagram, — 

Haec immatura a me lam frustra leguntur ; o. y. 

Some months later he furnished a translation, which is found by merely 
transposing the letters of the anagram and reads, " Cynthise figuras 3emu- 
latur Mater Amorum," meaning <' The Mother of the Loves (Venus) imi- 
tates the phases of Cynthia," i.e., of the moon. 


Galileo's ' 
of the 



Fig. 136. — Telescopic Appearances of Venus 

Fig. 136 represents the disk of the planet as seen at five 
points in its orbit. 1, 3, and 5 are taken respectively at 
superior conjunction, greatest elongation, and near inferior con- 
junction, while 2 and 4 are at intermediate points. Number 2 
is badly engraved, however; the sharp corners are impossible 
since the terminator is always a semi-ellipse (Sec. 205). 



The planet attains its maximum brightness thirty-six days 
before and after inferior conjunction, at a distance of about 38° 
or 39° from the sun, when its phase is like that of the moon 
about five days old. It then casts a strong shadow and, as 
already said, is easily visible by day with the naked eye. 

400. Albedo. — According to ZoUner, the albedo of the planet High albedo 
is 0.50, which is about three times that of the moon and almost o^^®^'^^- 
four times that of Mer- 
cury. It is, however, ex- 
ceeded by the reflecting 
power of the surface of 
Jupiter and Uranus, while 
that of Saturn appears to 
be about the same. 

This high reflecting 
power probably indicates 
that the surface is mostly 
covered with cloud, as 
few rocks or soils could 
match it in brightness. 

Lowell, however, de- 
nies the existence of any- 
thing like a continuous 
cloud veil such as has 
been generally supposed. 

401. Atmosphere of the Planet. — There is no question that 
this planet has an atmosphere of considerable density. 

When the planet is entering upon the sun's disk, or leaves 
it at a "transit," the portion of the disk outside the sun is 
encircled by a beautiful ring of light, due to the refraction, 
reflection, and dispersion of light by the planet's atmosphere 
(Fig. 137). If it were due solely to refraction, it would indi- 
cate that this atmosphere, according to the computations of 
Watson and others, must have an elevation of some ^b miles 

of Venus. 

Fig. 137. — Atmosphere of Venus as seen during 

the Transit of 1882 




Bright ring 
disk dvxe to 
and diffusion 
of light 
rather than 
to refrac- 

Density of 
the atmos- 
phere of 

Question as 
to presence 
of water 

light on 

and be considerably denser than our own; but this conclusion 
is very doubtful. 

When the planet is near the sun, about the time of inferior 
conjunction, the horns of its crescent extend notably beyond 
the diameter, and when very near the sun they can be seen, 
by carefully screening the object-glass of the telescope from 
the sunlight, to form a complete ring around the disk, as 
observed by Professor Lyman of New Haven and others in 
1860 and 1874. This phenomenon, which is unquestionable, 
has usually been ascribed to refraction ; but the observations 
of Russell at Princeton in 1898 showed that it must be due 
mainly to diffuse reflection of light by the planet's atmosphere, 
like that which causes our twilight, and that refraction proper 
plays only a very secondary part. 

If the ring were due to refraction, as by a lens, the widest 
and brightest part of it should be on the side of the planet 
mo%t distant from the sun, where the rays would be bent towards 
the observer, and not on the side next the sun, as is actually and 
conspicuously the case. On this side refracted rays are bent 
away from the observer and would not reach his eye, while 
reflected rays are thrown towards it. 

The same observations also cast doubt on the hitherto accepted 
conclusion as to the great density of the atmosphere, making it 
probable that it is somewhat rarer than our own, rather than 
much denser; and this might be expected, considering the 
planet's smaller mass and presumably higher temperature. 

The presence of water vapor in the planet's atmosphere has been 
announced by several of the earlier spectroscopic observers. The evidence, 
however, is hardly conclusive. 

Another curious phenomenon, not very satisfactorily explained as yet, is 
the occasional appearance of light on the unilluminated part of the planet's 
surface, making the whole disk visible, like the new moon in the old 
moon's arms. This light cannot be accounted for by any effect of sunlight, 
but must originate on the planet's surface or in her atmosphere. It recalls 
the aurora borealis of the earth and other electrical manifestations. 

thp: terrestrial and minor planets 


402. Surface Markings. — The surface of the planet is so Surface 
brilliant as seen in the telescope that it is very difficult to make "^^^^k^"^*^- 
out any markings upon it ; indeed, it is generally best in study- 
ing the surface to use a light shade glass. The disk is brightest 
at the limb, but the light fades off rapidly at the terminator, 
and over the surface there have been made out indistinct patches 
of less or greater brightness, as shown in Fig. 138, from draw- 
ings by Mascari made at the observatory on Mt. Etna in 1895, 
— an excellent representation of the planet's usual appearance. 
The darkest shadings may possibly be continents and oceans, 
dimly visible, though their comparative permanence with respect 
to the terminator 
makes this ques- 
tionable ; more 
probably they are 
purely atmos- 
pheric effects. 
But observations 
are as yet hardly 


very bright spots appear at the ends of the terminator, which Poiar caps, 
may possibly be polar caps like those of Mars, and, if so, show^ 
that the planet's axis must be nearly perpendicular to its orbit. 
On the terminator roughnesses and irregularities are sometimes 
seen which may perhaps be due to mountains, to some of which 
Schroter assigned extravagant elevations exceeding 20 miles. 

Lowell, in 1898, in opposition to all previous observers, reports Surface 
the discovery of permanent marking's consisting^ of rather narrow, ^^^^^^g^ 

/ ... according 

nearly straight, dark streaks, radiating like spokes from a sort to Lowell, 
of central hub. He describes them as fairly definite in outline, 
but dim^ as if seen through a luminous but unclouded atmos- 
phere of considerable depth ; and he goes so far as to give a 
map of the planet, w^ith names attached to some of the leading 

Liwii-, -iiwa I 

Fig. 138. — Venus 
After Mascari 



Rotation : 
planet keeps 
same face 
sun, like 

features. As yet, however, his observations want confirmation. 
Fig. 139 is from one of his drawings. 

403. Rotation, Position of Axis, etc The earlier observers, 

from the first Cassini in 1666 down to De Vico in 1841, assigned 
to the planet a rotation period of about 23^21™, — uncontra- 
dicted, it is true, but regarded with a good deal of distrust, 
because the observations were of spots extremely vague and 
indistinct and were not very accordant. 

Some highly respected authorities still accept this period, but 
the general opinion of astronomers now concurs with the con- 
clusion of Schiaparelli, who considers that his observations make 
it certain that the rotation must be very slow, and render it 

decision by 
the spectro- 

Fig. 189. — Venus 
From drawings of P. Lowell 

highly probable that Venus follows the example of Mercury in 
keeping the same face always towards the sun, having, therefore, 
a diurnal period of 225 days. This is confirmed by Perrotin at 
Nice, and by Lowell at the Flagstaff Observatory, and by several 
other observers. 

It is probable that the spectroscope will ultimately settle the 
question (though the observation will be very difficult) by show- 
ing, according to the Doppler principle (Sec. 254), how fast the 
eastern and western edges of the planet's disk respectively 
advance and recede ; the observation has already been attempted 
by Belopolsky at Pulkowa, and it is a little disconcerting that 
the result, though by no means decisive, rather favors the twenty- 
three-hour period. 


On the other hand, the planet shows no sensible oblateness, as No measur- 
it should if it had a day of the same length as the earth's ; if ^^^® oWate- 

•^ ° ' ness. 

that were the case, there should be a difference of nearly I" 
between the equatorial and polar diameters, which has never 
been observed. 

The inclination of the planet's equator cannot be exactly inclination 
determined, but it is almost certain that it must nearly coincide ^^ equator 

. . to orbit 

with the plane of its orbit. The old determination of De Vico, probably 
still found in many text-books, making the inclination 37°, is small, 
certainly erroneous. 

404. Question of Satellite. — No satellite has yet been dis- No satellite. 
covered, and it is certain that the planet has none of any con- 
siderable size. It is not impossible, however, that it may have 

some pygmy attendant, like those of Mars, since the great 
brilliance of the planet and its nearness to the sun would make 
the discovery of such a body extremely difficult. There have 
been in the past several announcements of a satellite ; but not 
one has been verified, and most of them were mistakes, since 
explained, either as observations of stars, or by reflections in 
the eyepiece of the observer's telescope. 

405. Transits. — Occasionally Venus passes between the Transits, 
earth and the sun at inferior conjunction and ''transits," 

or crosses, the disk of the sun from east to west as a round 
black spot, easily seen by the naked eye through a suitable 
shade glass. When the transit is central it occupies about 
eight hours, but when the track is near the edge of the disk 
it is correspondingly shortened. Since the transit can occur Transit 
only when the sun is within about 4° of the node, the phe- ^f°"^^^ 

«^ ' ^ June and 

nomenon is rare and can happen only within a day or two December, 
of the dates when the earth passes the nodes, viz., June 5 and 
December 7. 

The special interest of the transits lies in their availablity 
for the purpose of finding the parallax and distance of the sun, 
as first pointed out by Halley in 1679. 



Importance The earliest observed transit in 1639 was seen by two persons only 

of determin- (Horrocks and Crabtree, in England), but the four which have since 

ing the 
solar paral- 

and dates of 

Transits at 


come In 





occurred, in June, 1761 and 1769, and in December, 1874 and 1882, were 
extensively observed by scientific expeditions sent out by the different 
governments to all parts of the world where they were visible. The 
transits of 1769 and 1882 v/ere visible in this country. 

It is, however, hardly likely that so much trouble and expense will be 
hereafter expended upon observations of transits. Other methods of 
determining the solar parallax have been found to be more trustworthy. 

406. Recurrence and Dates of Transits. — Five synodic revolu- 
tions of Venus are very nearly equal to eight years, the differ- 
ence being little more than one day; and still more nearly, — 
in fact, almost exactly, — 243 years are equal to 152 synodic 
revolutions. If, then, we have a transit at any time, another 
may occur at the same node eight years earlier or later. Sixteen 
years before or after it will be impossible, and no other transit 

can then occur at the same node until after 
the lapse of 235 or 243 years, though a 
transit or pair of transits may, and usually 
will, occur at the other node in about half 
that time : thus, the next pair of transits 
of Venus will occur on June 8, 2004, and 
June 6, 2012. 

If the planet crosses the sun nearly cen- 
trally, the transit will be "solitary," ^.e., not 
accompanied by another eight years before 
or after. If, however, the track is more than 12' from the sun's 
center, it will be accompanied by another at eight years interval. 
At present transits come thus in pairs and have been doing so 
for several centuries; after a time this will cease to be the case, 
and they will become solitary for another long period. 

Fig. 140 shows the tracks of Venus across the sun's disk in 
1874 and 1882. 

Fig. 140. — Transit of 
Venus Tracks 



407. This planet, like Mercury and Venus, is prehistoric as Mars: data 
to its discovery. It is so conspicuous in color and brightness ^'e^atingto 
and in the extent and apparent capriciousness of its movement 

among the stars, that it could not have escaped the notice of the 
very earliest observers. 

Its mean distance from the sun is a little more than one and 
a half times that of the earth (141 500000 miles), and the eccen- 
tricity of its orbit is so considerable (0.093) that its radius vector 
varies more than 26 000000 miles. 

At opposition the planet's average distance from the earth is 
48 600000 miles. When opposition occurs near the planet's 
perihelion this distance is reduced to 35 500000 miles, while 
near aphelion it is over 61 000000. At superior conjunction 
the average distance from the earth is 234 000000. 

The apparent diameter and brilliancy of the planet vary enor- Enormous 
mously with those ofreat changes of distance. At a favorable ^^^"^^^^n of 

y ^ ^ _ ... . distance anc 

opposition (when the distance is at its minimum) the planet is brightness, 
more than fifty times as bright as at superior conjunction and 
fairly rivals Jupiter ; when most remote it is hardly as bright as 
the pole-star. 

The favorable oppositions occur always in the latter part of August (at Favorable 
which time the earth as seen from the sun passes the perihelion of the oppositions, 
planet) and at intervals of fifteen or seventeen years. The last such 
opposition was in 1892 ; the next will be in 1907. 

The inclination of the orbit is small, — 1° 51'. 

The planet's sidereal period is 687 days, or one year and ten 
and one-half months ; its synodic period is much the longest in 
the planetary system, being 780 days, or nearly two years and 
two months. During 710 of the 780 days it moves eastward, 
and during 70 retrogrades through an arc of 18°. 

408. Magnitude, Mass, etc. — The apparent diameter of the 
planet ranges from 3''. 6 at conjunction to 24''. 5 at a favorable 




and gravity. 



not dense. 

Question as 
to presence 
of water 

opposition. Its real diameter is very near 4200 miles. This 
makes its surface about two sevenths, and its volume one seventh, 
of the earth's. 

Its mass is a little less than one ninth of the earth's mass 
and is accurately determined by means of its satellites. Its 
density is 0.73, as compared with the earth's 0.73 and superficial 
gravity 0.38 ; a body which here weighs 100 pounds would have 
a weight of only 38 pounds on the surface of Mars. 

409. General Telescopic Aspect, Phases, Albedo, Atmosphere, 
etc. — When the planet is nearest the earth it is more favorably 
situated 1 for telescopic observation than any other heavenly 
body, the moon alone excepted. It then shows a ruddy disk 
which, with a power of 75, is as large as the moon. Since its 
orbit is outside the earth's, it never exhibits the crescent phases 
like Mercury and Venus ; but at quadrature it appears distinctly 
gibbous^ about like the moon three days from the full. 

Like Mercury, Venus, and the moon, its disk is brighter at 
the limb (i.e., at the circular edge) than at the center; but at 
the terminator^ or boundary between day and night on the 
planet's surface, there is a shading which, taken in connection 
with certain other phenomena, indicates the presence of an 

This atmosphere, however, contrary to opinions formerly held, 
is probably much less dense than that of the earth, the low den- 
sity being indicated by the infrequency of clouds and of other 
atmospheric phenomena familiar to us upon the earth, to say 
nothing of the fact that, since the planet's superficial gravity is 
less than two fifths of the force of gravity on the earth, a dense 
atmosphere would be impossible. 

More than twenty years ago Huggins, Janssen, and Vogel 
all reported the lines of water vapor in the spectrum of the 
planet's atmosphere ; but the observations of Campbell, at the 

1 Yenus at times comes nearer, but when nearest she is visible only by 
daylight, and shows only a very thin crescent. 


Lick Observatory in 1894, throw great doubt on their result 
and show that the water vapor, if present at all, is too small in 
amount to give decisive evidence of its presence. 

Zollner gives the albedo of Mars as 0.26, — just double that Albedo low. 
of Mercury, and much higher than that of the moon, but only 
about half that of Venus and the major planets. Near opposi- 
tion the brightness of the planet suddenly increases in the same 
way as that of the moon near the full (Sec. 210). 

410. Rotation, etc. — The spots upon the planet's disk enable Rotation 
us to determine its period of rotation with g-reat precision. Its P^^^^^'^ry 

^ . accurately 

sidereal day is found to be 24^^37^22^.67, with a probable error known. 
not to exceed one fiftieth of a second. This very exact deter- 
mination is effected by comparing drawings of the planet made 
by Huyghens and Hooke more than two hundred years ago with 
others made recently. 

The inclination of the planet's equator to the plane of its orbit Position of 
is very nearly 24° 50' (26° 21' to the ecliptic). So far, therefore, ^^^^" 
as depends upon that circumstance. Mars should have seasons 
substantially the same as our own, and certain phenomena of 
the planet's surface, soon to be described, make it evident that 
such is the case. 

The planet's rotation causes a slight but sensible flattening at Obiate- 
the poles, — about ^-^q- , according to the latest determinations. ^®^^ ^^''^ 

Much larger values, now certainly known to be erroneous, are found in 
the older text-books. 

411. Surface and Topography. — With even a small telescope. Surface 
not more than 3 or 4 inches in diameter, the planet is a very ^^^^i^gs. 
beautiful object, showing a surface diversified with markings 

dark and light, which, for the most part, are found to be perma- 
nent objects. Occasionally, however, for a few hours at a time, 
we see others of a temporary character, supposed to be clouds, 
since they for the time obliterate the permanent ones ; but these 
are surprisingly rare as compared with clouds upon the earth. 



Polar caps : 
question as 
to their 

spots called 
seas, but 
with doubt- 
ful pro- 

The permanent markings on the planet are broadly divisible 
into three classes. 

First, the white patches^ two of which are specially conspicuous 
near the planet's poles and are called the " polar caps." They 
are by many supposed to be masses of snow or ice, since they 
behave just as would be expected if such were the case. The 
northern one dwindles away during the northern summer, when 
the north pole is turned towards the sun, while the southern one 

grows rapidly 
larger ; and vice 
versa during the 
southern summer. 
But the prob- 
able low temper- 
at ure of the 
planet (Sec. 415) 
makes it at least 
doubtful whether 
the apparent 
"snow and ice" 
is really con- 
gealed water^ or 
some quite differ- 
ent substance. 
Second, patches of a bluish gray or greenish shade^ covering 
about three eighths of the planet's surface, until recently gener- 
ally supposed to be bodies of water, and therefore called " seas " 
and " oceans." But more recent observations, if they can be 
depended on, show a great variety of details within these areas, 
and such changes of appearance following the seasons of the 
planet, that this theory is no longer tenable, and they seem more 
likely to be regions covered with something like vegetation. 

Third, extensive regions of various shades of orange and yellow^ 
covering nearly five eighths of the surface, and interpreted as 

Fig. 141. — Mars 

Keeler, 1892 



land. These markings are, of course, best seen when near the Continents, 
center of the planet's disk ; near the limb they are lost in the 
brilliant light which there prevails, and at the terminator they 
fade out in the shade. 

Fig. 141, from drawings by Green of Madeira, and Fig. 142, from 
drawings by Keeler of the Lick Observatory, give an excellent idea of the 
planet's appearance as seen by most observers under good conditions. 

412. Recent Discoveries; the Canals and their Gemination. — 

In addition to these three classes of markings the Italian 
astronomer Schiaparelli, in 1877 and 1879, reported the discov- 
ery of a great number of fine straight lines, 
or " canals," as he called them, crossing the 
ruddy portions of the planet's disk in all 
directions, and in 1881 he announced that 
some of them become double at times. 

These new markings are faint and very 
difficult to see, and for several years there 
was a strong suspicion that he was misled by 
some illusion, — in respect to their "gemi- 
nation," at least, — which is still ascribed, 
by some very high authorities, to astigma- 
tism in the eye of the observer or bad 
focusing of his telescope. Still, the weight 
of evidence at present favors the reality of 
the phenomena which Schiaparelli describes. 
Many observers, both in Europe and the 
United States, have confirmed his results, 
and they are now generally accepted, although some of the best, 
armed with very powerful telescopes, still fail to see the canals 
as anything but the merest shadings, — not at all as shown in 
the drawings of Schiaparelli and Lowell. It appears that in the 
observation of these objects the power of the telescope is less 
important than steadiness of the air and keenness of the 

The canals. 

Their gemi- 

Fig. 142 
Green, 1878 

Canals diffi- 
cult to 
Question as 
to possible 



observer's vision. Nor are they usually best seen when Mars is 
nearest, but their visibility depends largely upon the season of 
the planet ; and this is especially the case with their " gemina- 
tion." Fig. 143, from one of Mr. Lowell's drawings made in 
1894, gives an idea of the extent and complexity of the canal 
system; but the reader must not suppose that in the telescope 
it stands out with any such conspicuousness. The figure shows 
also how some of the canals cross the so-called " seas " and dis- 
prove the propriety of the name. 

413. As to the real nature and office of the " canals " there is a wide 
difference of opinion, and it is very doubtful if their true explanation 
has yet been reached. Indeed, it is still quite possible that some of the 

Fig. 143. — Mars 
After Lowell 

peculiar phenomena reported are illusions, based on what the observers 

think they ought to see ; it is easy to be deceived in attempting to interpret 

intelligibly what is barely visible. 
Views of According to Flammarion, Lowell, and other zealous observers of the 

Flammarion planet, the polar caps are really snow sheets, which melt in the (Martian) 
and Lowell, spring and send the water towards the planet's equator over its nearly level 

plains (for no high mountains have yet been discovered there), obscuring 

for several weeks the well-known markings which are visible at other 


In Lowell's view the dark regions on the planet's surface are areas 

covered with some sort of vegetation, while the ruddy portions are barren 



I'll: 1,.^-SM,r.> Mv,mi Cik, 2 -Sn ,, ri^ \1 mok 

111,"'-'. . . r ,: 

deserts, intersected by the canals, which he believes to be really irrigating Office of 
watercourses ; and on account of their straightness, and some other charac- the canals, 
teristics, he is disposed to regard them as artificial. 

When the water reaches these canals vegetation springs up along their 
banks, and these belts of verdure are what we see with our telescopes, — 
not the narrow water channels themselves. 

Where the canals cross each other and the water supply is more abun- Vegetation 
dant there are dark round "lakes," as they have been called, which he and oases, 
interprets as oases. 

All of this theoretical explana- 
tion rests, however, upon the 
assumption that the planet's tem- 
perature is high enough to permit 
the existence of water in the 
liquid state, to say nothing of 
other difficulties. But whatever 
may be the explanation, there is 
no longer much doubt as to the 
existence of the canals, nor that 
they and other features of the sur- 
face undergo real changes with the 
progress of the planet's seasons. 

Their "gemination," however, 
still remains a mystery, nor is it 
entirely certain that it may not be 
a purely optical effect, as already 
intimated ; experiments made at 
Harvard College Observatory in 
1896, and later in France, point 
strongly in this direction. 

Certain changes on the surface 
of the planet are clearly connected with its seasons. This is, of course. Seasonal 
the case with the alternate growth and shrinkage of the polar caps, and changes. 
Flammarion and Lowell have reported others. Fig. 144, from the obser- 
vations of Lowell in 1894, shows their nature and amount. 

414. Maps of the Planet. — A number of maps of Mars have Maps of 
been constructed by different observers since the first was made ^^^^" 
by Maedler in 1830. Fig. 145 is reduced from one published 
in 1888 by Schiaparelli and shows most of his "canals" and 

-ll!:,r,„,, l-,^. 1 _|l| 

l-.t i; - ^iMs Trr.isiii 

Fig. 144. 

Seasonal Changes on Mars 















their " gemination." While there may be some doubt as to the 
accuracy of the minor details, it is probable that the main 
features of the planet's surface are substantially correct. 

The nomenclature, however, is in a very unsettled condition. 
Schiaparelli has taken his names mostly from ancient geography, 
while the English areographers,^ following the analogy of the 
lunar maps, have mainly used the names of astronomers who 
have contributed to our knowledge of the planet's surface. 

415. Temperature. — As to the temperature of Mars we have Temper- 
no certain knowledge at present. Unless the planet has some ^^^^'^ °* *^^^ 

p • ^ planet, 

unexplained sources of heat it ought to be very cold. a priori, 

Its distance from the sun reduces the intensity of solar radia- supposabiy 

tion upon its surface to less than half its value upon the earth, 

and its atmosphere cannot well be as dense as at the tops of our 

loftiest mountains. 

On the other hand, things look very much as if the poles 
were really snow-caip-ped, and as if liquid water and vegetable 
life were present in other regions. 

If so, we must suppose that the planet has sources of heat, Facts that 
external or internal, which are not yet explained; otherwise the ^^s^est 


polar "snow" must be something else than frozen water, as is sources 
perhaps not impossible. It is earnestly to be hoped, and may be ^^ ^®^*- 
expected, that before long we shall obtain some heat-measuring 
apparatus sufficiently delicate to decide whether the planet's 
surface is really intensely cold or reasonably warm, — for of 
course there are various conceivable hypotheses which might 
explain a high temperature at the surface of Mars. 

416. Satellites. — The planet has two satellites, discovered Satellites: 
by Hall, at Washingfton, in 1877. They are extremely small ^^^irdis- 

•^ ^ "^ '^ covery and 

and observable only with very large telescopes. The outer one, names. 
Deimos, is at a distance of 14600 miles from the planet's center 
and has a sidereal period of 30^18"^; while the inner one, Phobos, 

1 The Greek name of Mars is Ares ; hence, areography is the description of 
the surface of Mars. 



behavior of 


Question of 

Suggestion of 
in the system 
of canals. 

is at a distance of only 5800 miles and has a period of 7^39^", — 
less than one third of the planet's day. (This is the only case 
known of a satellite with a period shorter than the revolution 
of its primary.) Owing to this fact, it rises in the west, as seen 
from the planet's surface, and sets in the east, completing its 
strange backward diurnal revolution in about eleven hours. 
Deimos, on the other hand, rises in the east, but takes nearly 
132 hours in its diurnal circuit, which is njore than four of its 
months. Both the orbits are sensibly circular and lie very 
closely in the plane of the planet's equator. 

Micrometric measures of the diameter of such small objects 
are impossible, but, from photometric observations, Prof. E. C. 
Pickering, assuming that they have the same reflecting power 
as that of Mars itself, has estimated the diameter of Phobos as 
about 7 miles and that of Deimos as 5 or 6. Mr. Lowell, how- 
ever, from his observations of 1894, deduces considerably larger 
values, viz., 10 miles for Deimos and 36 for Phobos. If this is 
correct, Phobos, seen in the zenith from the point on the planet's 
surface directly beneath him, would appear somewhat larger than 
the moon, but only about half as bright. Deimos would be no 
brighter than Venus. 

417. Habitability of Mars. — As to this question we can only 
say that, different as must be the conditions on Mars from those 
prevailing on the earth, they differ less from ours than those on 
any other heavenly body observable with our present telescopes; 
and if life, such as we know it upon the earth, can exist on any 
of the planets, Mars is the one. If we could waive the ques- 
tion of temperature and assume, with Flammarion and others, 
that the polar caps really consist of frozen water, then it would 
become extremely probable that the growth of vegetation is the 
explanation of many of the phenomena actually observed. 

Mr. Lowell goes further and argues the presence of intelligent 
beings, possessed of high engineering skill, from the apparent 
" accuracy " with which the " canals " seem to be laid out in a 


well-planned system of irrigation. But at present, and until 
the temperature problem is solved, such speculations appear 
rather premature; and as to the establishment of communica- 
tion with the hypothetical inhabitants, the idea, in the present 
state of human arts at least, is simply chimerical. 


418. The " asteroids," or minor planets, are a host of small 
bodies circulating around the sun between the orbits of Mars and 
Jupiter. The name " asteroid," i.e.^ " starlike," was suggested 
by Sir William Herschel early in the century, as indicating that, 
though really planets,' they appear like stars. 

Kepler had noticed the wide gap between Mars and Jupiter The aster- 
and had tried to account for it, though unsuccessfully, and ^^^^^ takmg 
when Bode's Law (Sec. 349) was published in 1772 the impres- of a single 
sion became very strong that there must be a missing planet in planet m 

. . region indi- 

tne vacant space, — an impression greatly strengthened by the ^ated by 
discovery of Uranus in 1781, at a distance almost precisely Bode's Law, 
corresponding to that law. An association of twenty-four 
astronomers, mostly German, was formed to look for the miss- 
ing planet, but failed to find one after a dozen years of search, 
and the first discovery was made by the Sicilian astronomer, 
Piazzi, who was then engaged in forming his extensive cata- 
logue of stars. 

On the first night of the nineteenth century (Jan. 1, 1801) Discovery 
he observed a small star where there had been no star a few ^^ ^^^^'^ ^^ 


days earlier, and the next day it had obviously moved, and it 
continued to move. He named the new planet Ceres^ after the 
tutelary divinity of the island, and observed it carefully for 
several weeks, until he was taken ill; but before he recovered 
the planet was lost in the evening twilight. It was redis- 
covered at the close of the next year by means of Gauss' 
calculations. (See Sec. 365.) 



of Pallas, 
Juno, and 

of Astrsea. 

Over five 



by numbers 
and names. 

Method of 
search Avith 


In 1802, while searching for Ceres, Pallas was discovered by 
Olbers. Juno was found by Harding in 1804, and in 1807 
Olbers, who had broached the theory that these new bodies 
were fragments of an exploded planet, discovered Vesta, the 
only one ever visible to the naked eye. The search was kept up 
for several years longer without success, because those engaged 
in it did not look for small enough objects. 

The fifth asteroid, Astrcea^ was discovered in 1845 by Hencke, 
an amateur, who had resumed the search afresh by studying 
the smaller stars and after fifteen years of fruitless labor was 
rewarded by the new discovery. In 1847 three more were found, 
and not a year has passed since then without the discovery of 
from one to thirty. 

At present (January, 1902) the number known exceeds five 
hundred, and for the past ten years has been increasing with 
great rapidity. 

They are all designated by numbers ; i.e.^ each one receives 
a number after having been observed a sufficient number of 
times to determine its orbit, which usually happens soon after 
its discovery. Most of them also have names, usually mytho- 
logical, and feminine for all but Eros ; but it has already become 
a pretty serious matter to find names for all the new discoveries, 
and the custom may be given up before long. 

419. Method of Search. ^ — Formerly the asteroid hunter con- 
ducted his operations by making special telescopic star charts 
of regions near the ecliptic, and from time to time comparing 
the chart with the heavens. If an interloper appeared on the 
chart, a few hours' watching would decide whether it moved or 
not, i.e., whether it was a planet or merely a variable star. The 
work, especially that of chart making, was very laborious. 

In 1891 a new method was introduced by Dr. Max Wolf of 
Heidelberg. A camera with a wide-angle lens of several inches 
aperture is mounted equatorially and moved by clockwork; 
with this photographs are made of portions of the sky from 5° 

THE terrp:strial and minor planets 


to 10° in diameter. On the negative the stars, if the clock- 
work runs correctly, show as small black dots, but a planet, if 
present, will move among the stars during the two or three 
hours of exposure, and its image will be a streak instead of a 
dot, and so recognizable at once. 

Fig. 146 is a direct reproduction of the plate on which Dr. Wolf dis- 
covered planet 1892, V (Giidrun, (328) ), the " trail " of which, due to about 
two hours motion, is shown exactly in the center of the cut. The first 
planet discovered by this method, in December, 1891, AVoU has named 
"Brucia," (323), in honor of 
the late Miss Catherine W. 
Bruce of New York, who pro- 
vided the funds for his camera 
and its mounting. 

It has happened several 
times that more than one 
planet is found on the 
negative ; in one instance 
as many as five, three of 
which were new, and in 
another (on a plate made 
at Harvard) no less than 
seven. Already, during 
the past ten years, nearly 
two hundred have been thus discovered, almost all by Wolf of 
Heidelberg and Charlois of Nice, though a few others have 

Great care is necessary to be sure that the objects discovered are really 
7i€w. There are a number of the older ones which, not having been observed 
for many years, are now adrift and practically lost, and are likely to be 
rediscovered at any time. Several of them indeed have been already 
picked up by the new method. 

Since 1892 the newly discovered bodies, while awaiting the final num- 
bers (and perhaps a name), are provisionally designated by letters, as AM, 
DQ, etc. ; they are now (1902) far along in the H's. 

Fig. 14G. — Wolf's Discovery of (328) 
"Gudruh," 1892 

of number. 



A full list, brought down to date as nearly as possible, is published 
every year in the Annuaire du Bureau des Longitudes, Paris, giving their 
number, name, date of discovery, and the elements of their orbits. 


and period : 






three to nine 


mean dis- 
tance of 
group about 

Large incli- 
nations and 


420. Mean Distance and Period. — The mean distances of the 
different asteroids from the sun differ widely, and their periods 
correspond. Excepting Eros, the nearest to the sun so far as 
yet determined is Adalberta, (330), its mean distance being about 
2.09 (194 000000 miles) and its period three years and three 
days. Thule, (279), is the most remote, with a distance of 4.30, 
or 400 000000 miles, and a period only a month less than nine 

The mean distances are not distributed at all uniformly 
through their range, but there are several marked gaps, doubt- 
less due to the action of Jupiter, since they come just where 
the period of the asteroid would be exactly commensurable 
with that of the great planet, i.e., |-, 1, |, or J of Jupiter's 
period. The distances are grouped most densely about 2.8, 
which Bode's Law would indicate as that of the "missing" 
planet; but the average mean distance comes out somewhat 
smaller, about 2.65 (246 500000 miles), corresponding to a 
period of about four and one-third years. 

421. Inclinations and Eccentricities. — These average much 
greater than for the principal planets. The mean inclina- 
tion of the asteroid orbits to the ecliptic is about 8°. The 
orbit of Pallas, (2), is inclined 35°, and seven or eight others 
exceed 25°. 

The eccentricity is also very large in some cases. For 
^thra,^ (132), and Andromache, (175), it is fairly cometary, 
exceeding 0.35, and there are a dozen others above 0.30. 

1 A planet very recently discovered (October, 1901) but not yet numbered 
proves to have an eccentricity greater than even that of ^thra, exceeding 0.38. 


The orbits so cross and interlink that if they were material 
hoops or rings the lifting of one would take all the others with 
it, and that of Mars also, caught up by that of Eros. 

422. Diameter, Surface, etc These bodies are so small that Diameters 

micrometrical measurements, even of the largest, are extremely ^^^' ^^^'"^^ 

' , . , . part too 

difficult, and of the smaller ones impossible. Since 1890, small for 
however, Barnard, with the Lick and Yerkes telescopes, has "measure- 

111 611 1 

obtained measures of the disks of the four brightest and presum- 
ably largest, with the following rather surprising results, viz. : 
Ceres, 488 miles; Pallas, 304; Vesta, 248; Juno, 118. The Barnard's 
surprise consists in the fact that Vesta, which is fully twice as ^'f^^^^^ ^^^ . 
bright as Ceres, should have a diameter only half as great, paiias, 
showing a wide difference of albedo. Mliller of Potsdam, Vesta, and 
accepting Barnard's diameters, finds for Ceres from his photo- 
metric observations an albedo about the same as that of Mercury 
(0.13), while that of Vesta is put at 0.72, — higher than that of 
any other planet, and nearly equal to that of writing-paper. 

As to the other asteroids, probably no one of them has a 
diameter as great as 100 miles, and the smaller ones, such as 
those which are now being discovered, are mostly of the thir- 
teenth and fourteenth magnitude, so small that they cannot be 
seen (though easily photographed) with a telescope of much less 
than 12 inches aperture and cannot be more than 10 or 15 miles 
in diameter, — mere "mountains broke loose," with a surface 
area no more extensive than some western farms. 

423. Mass and Density. — On these points we have no abso- Mass of 
lute knowledge ; but if we assume that the density is about the ^®^®^ ^f ^'' 

haps 5800 of 

same as that of the planet Mars (seventy-three per cent of the earth's. 
density of the earth), which is probably an overestimate, Ceres 
would have a mass of about -^-^-q-q tliat of the earth, and the force 
of gravity at her surface would be about -^^ of gravity here. 

A stone would descend only about 8^ inches in the first second of its Force of 
fall, and the " parabolic velocity " at the planet's surface would be about gravity on 
1900 feet a second (Sec. 319), — a rifle bullet shot from the planet would 



Total mass 
of group 
less than 
I of earth's, 
perhaps less 
than jhjs. 


never return. For a ^^lanet 10 miles in diameter of the same density, the 
critical velocity would be only 38 feet a second, so that if the hypothetical 
dweller on one of these " planetules," as Miss Gierke calls them, should 
throw away a stone, it would never come back, but would become an inde- 
pendent planet. 

It is, however, possible from the perturbations which the 
asteroids produce (or rather do not produce) on Mars to esti- 
mate, for the aggregate mass of the flock, a limit which it 
cannot exceed, — including the presumably undiscovered multi- 
tude as well as the five hundred now known. Leverrier found 
long ago that tlie total mass could not be as great as one quarter 
the mass of the earth ; and a much more recent computation by 
Ravene in 1896 puts it below one per cent. The united mass 
of all thus far discovered would make but a small fraction of 
this one per cent, — certainly not over j-^-q-q of the mass of the 

424. The number not yet discovered is probably enormous, 
though it is practically certain that nearly all that exceed 40 or 
50 miles in diameter are already in our catalogue. How long 
it will be considered worth while to search for new ones is 
doubtful, as it is quite certain that the computers will not 
continue to follow by calculation the motions of any except 
such as possess peculiar interest for their size or some other 

Theories as 
to oris:in. 

An asteroid is much more difficult to observe than a large planet, and 
immensely more troublesome to follow by calculation, because of the great 
perturbations to which it is exposed from Jupiter's attraction. One little 
family of these bodies, twenty-two in number, which were discovered by 
Professor Watson of Ann Arbor, is, however, " endowed " with a fund 
which he left in his will to pay for the calculations necessary to keep them 
from getting lost. 

425. Origin. — As to this we can only speculate. It is hardly 
possible to doubt that this swarm of little rocks in some way 
represents a single planet of the terrestrial group. 


A generally accepted view is that the material, which, accord- 
ing to the nebular hypothesis, once formed a ring or rings like 
those of Saturn, either continuous or of separate pieces, — matter 
which ought to have collected to make a single planet, — has A ring dis- 
f ailed to be so concentrated ; and the failure is ascribed to the ^p^*"^^ '^^ 

attraction of 

perturbations produced by its neighbor, the giant Jupiter, whose jnpiter. 
powerful attraction is supposed to have disintegrated the ring, 
or at least prevented the union of the separate parts, and thus 
stopped the development of a normal planet. 

Another view is that the asteroids may be fragments of an An exploded 
exploded planet. If so, there must have been not one but many p^''^"^^- 
explosions ; first of the original, and then of the separate pieces 
in different portions of their orbits. It is demonstrable that no 
single explosion could account for the present tangle of orbits. 

426. The Planet Eros. — This little planet, insignificant in Eros a 
size but of great astronomical interest, and already several times doubtful 
referred to, should probably be regarded as a member of the the asteroid 
asteroid family. It has, however, an orbit so much smaller than family. 
any other asteroid that the discoverer claims for it a status of 

its own. 

It was discovered in August, 1898, photographically, by Witt its dis- 
of Berlin, and at once attracted notice by the rapidity of its ^^^'«^^T- 
motion. After a preliminary calculation of its orbit had been 
made by Dr. Chandler, so that its place could be approximately 
computed for dates in the past, its trail was found on a con- 
siderable number of photographic plates made at Harvard Col- 
lege Observatory during several years preceding (1893, 1894, 
and 1896); and this rendered it possible at once to compute 
a very accurate orbit. 

427. Orbit of Eros. — Its mean distance from the sun is only its orbital 
1.46 (135 500000 miles). Its sidereal period is 643 days, its pecuiiari- 
synodic 845 days. The eccentricity of its orbit is 0.^2, which 
makes the aphelion distance 165 500000 miles (well outside the 

orbit of Mars and well within the asteroid region), while its 



Its occa- 
sional close 
approach to 
the earth. 
as a means 
of determin- 
ing solar 

Next oppor- 
tunity in 

range of 


variations of 

perihelion distance (105 300000 miles) is only a little more 
than 12 000000 miles greater than the mean distance of the 
earth from the sun. Its orbit is shown in Fig. 125. 

The inclination of its orbit is 11°, and this, combined with 
the fact that tlie perihelion of the planet's orbit nearly faces 
that of the earth, makes the least possible distance between the 
earth and Eros about IS 500000 miles. This is only a little 
more than half the least distance of Venus, and it gives the 
planet immense importance from an astronomical point of view, 
since observations made at such a time of close approach will 
furnish a far more precise determination of the solar parallax 
and astronomical unit than any other method known. 

Unfortunately, these close oppositions are rare ; one occurred 
in 1894, and another such opportunity will not occur again 
until 1931. In the winter of 1900-01 the conditions were 
better than they will be until then, the planet having come 
within about 30 000000 miles of the earth. An extensive series 
of observations, both visual and photographic, was made, par- 
ticipated in by all the leading observatories of the world. The 
mass of material accumulated is such that it will probably be 
two or three years at least before the results can be fully 
worked out. 

428. Eros itself. — The planet is small, probably not more 
than 15 or 20 miles in diameter, though this is merely an esti- 
mate. On account of the enormous variation in its distance 
from the earth (from 13 500000 miles to 260 000000), its bright- 
ness when nearest us is nearly four hundred times as great as 
when remotest. Near aphelion it is observed, if at all, only 
with the very largest telescopes ; when nearest, in 1931, it will 
for a few days, perhaps, become visible even to the naked eye. 

A very remarkable thing is the apparent periodic variation in 
its brightness observed during the early winter and spring of 
1901, — shown also in some of the Harvard photographs of 
1894 and 1896. At certain times the variation was very 


striking, the planet in February and March, 1901, being at the 
maximum fully three times as bright as at the minimum, only 
two and one-half hours later. At other times the variation 
disappeared entirely, as in May, 1901. The period of varia- 
tion, which is 5^16"", gives some evidence of two unequal half 
periods, one of 2^'25"' and the other of 2^51™, but this is not 
yet certain. 

The most natural explanation of the variation, as ah'eady mentioned Explained 

(Sec. 383), is that it is caused by the axial rotation of the planet, which is ^^y rotation 

supposed to have liaht and dark markings on its surface. If these are ^ ^ ^^^ ^ 

, . spliere or l)y 

arranged something like the continents on the earth (continents light, ^j^^ ;^.gvo- 

oceans dark), the variation of light would be about as observed when we lution of 

are in the plane of the planet's equator, and would cease when the planet's two bodies 

pole is directed towards us. ^^^~ 

Another explanation, preferred by the French astronomer Andre, is that . 
the planet is double, " a pair of twins," consisting of two bodies revolving around 
around each other almost in contact, in an oval orbit, and with a period of eacli other. 
5iqQm^ When we are in the plane of the orbit occultations occur twice in 
every revolution, one of the bodies eclipsing the other ; but on account of 
the eccentricity of the orbit these eclipses are not at equal intervals. 

To account for the greatness of the light change Andr6 further supposes 
that the bodies are egg-sJiaped, on account of their mutual tidal action, so 
that when seen sidewise they present three times as much surface as when 
seen " end on," one behind the other. When we are very much above or 
below the orbit-plane, so that the bodies pass each other without eclipse, 
the variation of light would cease. It remains to be seen what future 
observations may show as to the rival theories. 

429. Intramercurial Planets. — It is not impossible that there Possible 
is a considerable quantity of matter circulating around the sun ^tramer- 
inside the orbit of Mercury. This has been believed to be indi- planets, 
cated by the otherwise unexplained advance of the perihelion 
of Mercury's orbit, but the investigations of Newcomb render 
very doubtful the validity of such an explanation, since the 
nodes of the planet's orbit are not affected as they would be on 
that hypothesis. It has been somewhat persistently supposed 
that this intramercurial matter is concentrated into one, or 



of Vulcan 

possibly two, planets of considerable size, and such a planet 
has several times been reported as discovered, natably in 1857 
(when it was even named "Vulcan"), and again in 1878. We 
can only say that the supposed discoveries have never been con- 
firmed, and the careful observations during total solar eclipses 
during the past twenty years make it practically certain that 
there is no " Vulcan," z.e., no single considerable planet. Per- 
haps, however, there may be an intramercurial family of aster- 
oids. If so, they must be very small or some of them would 
certainly have been found during the eclipses ; and, if as large 
as 100 or 200 miles in diameter, some of them would probably 
have been caught crossing the sun's disk. 

Attempts to 
find intra- 
planets by 
during solar 

The zodiacal 
light: its 
and when 
best seen. 

The Gegen- 

An attempt was made to detect any existing body of this kind during 
the eclipses of 1900 and 1901 by means of photography, with a photo- 
graphic lens of long focus arranged to throw the image of an extensive 
region of the sky on a collection of photographic plates arranged upon a 
suitable frame behind the lens. The attempt failed on both occasions. 

430. The Zodiacal Light. — This is a faint pyramid of light, 
for the most part less luminous than the Milky Way, extending 
from the sun both east and west along the ecliptic. In northern 
latitudes it is best seen in the evening during the months of 
February and March ; in the morning, in October and November. 
Its summit is sometimes as far as 90° from the sun, and in the 
tropics it is said to be sometimes visible at midnight as a com- 
plete belt extending clear across the heavens. 

Opposite to the sun there is a slightly brighter patch 10° 
or 20° in diameter, called the "Gegenschein," or "counter 
glow." This, and indeed the whole phenomenon, is so faint 
that it can be well seen only when the observer is where there 
is no interference from artificial lights. Even the presence of 
one of the brighter planets greatly embarrasses the observation. 
The region near the sun is fairly bright, it is true, but is always 
more or less immersed in the twilight. 


The spectrum is a simple continuous one, witJiout perceptible The spec- 
liiies or marking of any kind. We emphasize this because it |'"^".'^ *^^ ^^'^ 


has often been erroneously reported that it shows the bright light. 
yellow line which characterizes the spectrum of the aurora 

The most probable explanation of the zodiacal light is that Probable 
it is due to reflection of sunlight from myriads of small particles ^^Pj^"^*!^" 
revolving around the sun in a comparatively thin, flat sheet or zodiacal 
ring (something like Saturn's ring), which extends far beyond ^^^^^^ ^^ ^ 
the orbit of the earth, and perhaps even to that of Mars. nng. 

Near the sun the particles are supposed to be more numerous 
than elsewhere, as well as more brilliantly illuminated, so that 
although less than half the sunlit surface of each is visible to 
us, yet on the whole the total sum of brightness is greater than 

As for the Gegenschein, this may be accounted for by sup- 
posing that the particles nearly opposite the sun "flash out" in 
the same way that the moon does at the full. 

It has been attempted to explain the zodiacal light as due to a ring 
of meteoric particles revolving around the earth ; but in that case the 
Gegenschein would be replaced by a dark spot caused by the shadow of 
the earth, unless indeed the ring had a very improbable diameter of many 
million miles. As to the size of the particles, we have no direct evidence, 
though from the analogy of shooting-stars it is likely that they are very 
small, perhaps not larger than pinheads. 

But it cannot be said that the problem is completely solved. There 
are serious difficulties with every theory yet proposed. 



Jupiter : its Satellite System ; the Equation of Light, and the Distance of the Sun — 
Saturn: its Rings and Satellites — Uranus: its Discovery, Peculiarities, and 
Satellites — Neptune: its Discovery, Peculiarities, and Satellite 




Jupiter: JuPiTER, the nearest of the major planets, stands next to 

Its conspicu- Yenus in the order of brilliance among the heavenly bodies, 
being five or six times as bright as Sirius, the most brilliant of 
the stars, and decidedly superior to Mars, even when Mars is 
nearest. It is not, like Venus, confined to the twilight sky, 
but at the time of opposition dominates the heavens all night 

431. Its orbit presents no marked peculiarities. The mean 
distance of the planet from the sun is a little more than five 
astronomical units (483 000000 miles), and the eccentricity of 
the orbit is not quite ^V' ^^ ^^^^^ ^^^ distance from the sun varies 
about 42 000000 miles between perihelion and aphelion. 

At an average opposition the planet's distance from the earth 
is about 390 000000 miles, while at conjunction it is about 
580 000000 ; but it may come as near to us as 370 000000 and 
may recede to a distance of nearly 600 000000. 

The sidereal period is 11.86 years, and the synodic period is 
399 days (a figure easily remembered), a little more than a year 
and a month. 

432. Diameter, Mass, Density, etc. — The planet's apparent 
diameter ranges from 50" to 32'', according to its distance from 
the earth. The disk, however, is distinctly oval, the equatorial 


surface, and 

times that 
of the earth. 


diameter being nearly 90000 miles, while the j^olar diameter is 
84200. The mean diameter ( ^^^^ \ (see Sec. 139) is 88000 
miles, or a little over eleven times that of the earth. 

These values are from the recent measures of Barnard and See, and are 
notably larger than those determined by earlier observers with a double- 
image micrometer and given in the table in the Appendix. Very likely the 
truth may lie intermediate. 

The oblateness is y^g, — very much greater than that of any 
other planet, Saturn excepted. 

Its surface is 122, and its volume or bulk 1355, times that 
of the earth. It is by far the largest of all planets, — larger, in 
fact, than all the rest united. 

Its mass is very accurately known, both by means of its its mass 317 
satellites and by the perturbations which it produces upon cer- 
tain asteroids. It is 77-7^-^ of the sun's mass, or about 317 

1048. o5 

times that of the earth. 

Comparing this with its volume, we find its mean density 
to be 0.23, z.e., less than one fourth the density of the earth its density 
and a little less than that of the sun. Its surface qravity is ^^^^ surface 

•^ '^ ^ gravity. 

about two and two-thirds times that of the earth, but varies 
nearly twenty per cent between the equator and poles of the 

433. General Telescopic Aspect, Albedo, etc. — In even a small General 
telescope the planet is a fine object, since a magnifying power ^^P®^*^- 
of only 60 makes its apparent diameter, even when remotest, 
equal to that of the moon. With a large instrument and mag- 
nifying power of 300 or 400 the disk is covered with an infinite 
variety of detail, interesting in outline, rich in color,- — mostly 
reds and brown, with here and there an olive-green, — and these 
details change continually as the planet turns on its axis. 

For the most part, the markings are arranged in "belts" The belts, 
parallel to the planet's equator, as shown in Fig. 147. The 



left-hand one of the two larger figures is from a drawing by 
Trouvelot (1870), and the other from one by Vogel (1880). 
The smaller figure below represents the planet's ordinary 
appearance in a 3-incli telescope. Fig. 148 is from a beauti- 
ful drawing by Keeler, made in 1889, which still continues to 

albedo, — 

Fig. 147. — Telescopic Views of Jupiter 

be an excellent representation of the planet's aspect. Near the 
limb of the planet the light is less brilliant than in the center 
of the disk, and the belts there fade out. 

The planet shows no perceptible phases^ but at quadrature 
the edge which is turned away from the sun is sensibly darker 
than the other. 

According to Zollner, the mean albedo of the planet is 0.62, 
which is very high, that of white paper being 0.78. The ques- 
tion has been raised whether Jupiter is not to some extent self- 
luminous, but there is no proof, and little probability, that such 
is the case. 



Fig. 148.— Jupiter 
After drawings by Keeler, at Lick Observatory 



The planet's 

of the 
Shadings in 
the red and 

period about 
for different 
classes of 

434. Atmosphere and Spectrum. — The planet's atmosphere 
must be very extensive. The forms visible with the telescope 
are nearly all evidently " atmospheric," — i.e.^ like clouds, — as 
is obvious from their rapid changes, though Professor Hough 
considers that we see the pasty, semi-liquid surface of the globe 
itself at times. The low mean density of the planet makes it, 
however, very doubtful whether there is anything solid about it 
anywhere, — ^ whether it is anything more than a ball of fluid, 
overlaid by cloud and vapor. 

The spectrum of the planet differs less from that of mere 
reflected sunlight than might have been expected, showing that 
the light is not obliged to penetrate the atmosphere to any great 
depth before it is reflected towards us from the clouds. There 
are, however, faint shadings in the red and orange parts of the 
spectrum that are probably due to some unidentified constituent 
of the planet's atmosphere; they seem to be identical in position 
with certain bands which are intense in the spectra of Uranus 
and Neptune. 

435. Rotation. — Jupiter rotates on its axis more swiftly than 
any other planet, — in about 9^^55"\ The time can be given 
only approximately; not because it is difficult to find, and to 
observe with accuracy, well-defined objects on the disk, but 
because different results are obtained from different spots, rang- 
ing all the way from 9^50"^ for certain small bright spots to 
9^56^"" for others of a different character. Well-marked features 
near each other on the planet's surface often drift by each other, 
sometimes at the rate of from 200 to 400 miles an hour. 

On the whole, spots near the equator usually show a shorter 
period than those in higher latitudes, but there are numerous 
exceptions. There is no such regular difference as on the sun, 
but there apparently are a number of different zones, each with 
its own rate of rotation, and one or two of the swiftest are not 
near the equator ; neither are the two hemispheres, the northern 
and southern, alike in their behavior. 


The plane of rotation nearly coincides with that of the orbit, Plane of 
the inclination being^ only 3°, so that there can be no well- ^'^^^^^^^^ 

^ "^ nearly in 

marked seasons on the planet due to causes such as produce plane of 
our own seasons. °^^^^- 

436. Physical Condition; the << Great Red Spot.'' — The con- 
dition of the planet is obviously very different from that of the 

earth or Mars. No permanent markings are found upon the Noperma- 
disk, though there are some which may be called at least sub- "^nt surface 

^ _ _ ^ ^ markings. 

permanent^ persisting for years with only slight apparent change. 

The most remarkable instance of such a marking is the great The great 
red spot, shown in Figs. 147 and 148. It was first noted in ^"^dspot. 
1878, was extremely conspicuous for several years, and then 
gradually faded away, slightly changing its form and becoming 
rounder; even yet (1901), while hardly visible - itself , the place 
which it occupies is clearly marked by the "bed" it has hol- 
lowed out in the great southern belt. In its prime it was about 
30000 miles long by about 7000 wide. 

Were it not that during the first six or seven years of its 
visibility it lengthened its rotation period by about six seconds 
(from 9^55^35® to 9'^55"41^), we might suppose it permanently 
attached to a solid nucleus below; but this change of rotation 
means that, relative to its position in 1878, the spot must have 
traveled completely around the nucleus of the planet in the six 
years, unless the nucleus itself changed its own period to the 
same extent, and that without affecting the motions of the other 
spots and markings. 

No really satisfactory explanation of the spot and its strange 
behavior has yet been found. 

437. Temperature. — Many things about the planet indicate Temper- 

a probable high teynperature, as, for instance, the abundance of ^^^^^ woh- 
clouds and the rapidity of their motions and transformations. The planet 
which almost certainly indicate a rapid exchange of matter and ^ ^®"^^- 
a vigorous vertical circulation between the surface and the 
underlying nucleus, if there is one. To maintain such an 




The five 
satellites of 
Jupiter ; 
their dis- 

The fifth 

Data relat- 
ing to the 

Third satel- 
lite the 

of the fourth 
satellite : 
very dark 

ebullition requires a continuous supply of heat, and since on 
Jupiter the solar light and heat are only 2^^ as intense as here, we 
are forced to conclude that it gets very little of its heat from the 
sun, but is probably hot on its own account, and for the same 
reason that the sun is hot, i.e., as the result of a process of con- 
densation. In short, it appears very probable, as has been inti- 
mated before, that the planet is a sort of " semi-sun," — hot, 
though not so hot as to be sensibly self-luminous. 

438. Satellites. — Jupiter has five satellites, four of them so 
large as to be seen easily with a common opera-glass. These 
were in a sense the first heavenly bodies ever " discovered," 
having been found by Galileo in January, 1610, with his newly 
invented telescope. The fifth satellite, discovered by Barnard 
at the Lick Observatory in 1892, is, on the other hand, extremely 
small and visible only in the most powerful instruments. 

It is nearest to the planet, its distance from the center of Jupi- 
ter being only 112500 miles and its sidereal period 11^57^'^.4. 
Its diameter probably does not exceed 100 miles. 

The old satellites, though more remote, are still usually known 
as the first, second, etc., in the order of their distance from the 
planet. Their distances range from 262000 to 1 169000 miles 
and their sidereal periods from forty-two hours to sixteen and 
two-thirds days. Their orbits are almost perfectly circular 
and lie very nearly in the plane of the planet's equator. The 
third satellite is much the largest, having a diameter of about 
3600 miles, w^hile the others are between 2000 and 3000, — all 
of them larger than our moon, though much less massive. 

For some reason, the fourth satellite is a very dark-com- 
plexioned body, so that when it crosses the planet's disk it 
looks like a black spot, hardly distinguishable from its own 
shadow; the others under similar circumstances appear bright, 
dark, or are invisible, according to the brightness of the part of 
the planet which happens to form the background. With ver}^ 
powerful instruments spots are sometimes visible on their 


surfaces, and there are variations in their brightness; W. H. 
Pickering, Douglass, and some other observers have also reported 
periodic irregularities in their forms, as if they were cloudlike 
in constitution. 

In the case of the fourth satellite the regularity in the changes 
of brightness indicates that it follows the example of our moon 
in always keeping the same face towards the planet, and the Keeps same 
observations of Douglass at Flagstaff, in 1897, of spots upon the ^^^® ^^ 
surfaces of the third and fourth satellites also indicate a rotation dnrino- its 
agreeing with their orbital periods far within the limits of error notation, 
to be expected in such observations. It may be considered prac- 
tically certain that both these satellites behave like our moon. 

The four satellites of Galileo have names also : viz., lo, Eiiropa, Gany- Names of 
mede, and Callisto, — lo being the nearest to the planet. But these names Gralilean 
are seldom used. 


439. Eclipses and Transits. — The orbits of the satellites are Eclipses and 
so nearly in the plane of the planet's orbit that with the excep- ^^'^^^^^^ ^^ 

. . . the satel- 

tion of the fourth, which at certain times escapes, they are ntes. 
eclipsed at every revolution, and also cross the planet's disk at 
every conjunction. 

When the planet is either at opposition or conjunction the 
shadow, of course, is directly behind it, and we cannot see the 
eclipse at all. At other times we ordinarily see only the begin- 
ning or the end ; but when the planet is at or near quadrature 
the shadow projects so far to one side that the whole eclipse of 
every satellite, except the first, takes place clear of the disk. 

An eclipse is a gradual phenomenon, the satellite disappear- The phe- 
ing by becoming slowly fainter and fainter as it plunges into ]^,*^"^^^^^ 
the shadow, and reappearing in the same leisurely way. 

Two important uses have been made of these eclipses : they Their use in 
have been employed for the determination of longitude, and ''^^^^■"^^'"y- 
they furnish the means of ascertaining the time required hy light 
to traverse the space between the earth and the s\in. 



The equa- 
tion of 
light: its 
the time 
occupied by 
light in 
from sun 
to earth. 

of the pro- 
motion of 

and acceler- 
ation of the 
eclipses as 
the distance 
of the earth 
from the 

440. The Equation of Light — When we obsei-ve a celestial 
body we see it, not as it is at the moment of observation, but as 
it was at the moment when the light which we see left it. If 
we know its distance in astronomical units, and know how long 
light takes to traverse that unit, we can at once correct our 
observation by simply dating it hack to the time when the light 
started from the object. 

The necessary correction is called the Equation of Light, and 
the time required hy light to traverse the astronomical unit of 
distance is the Constant of the light-equation (not quite five 
hundred seconds, as we shall see). 

It was in 1675 that Roemer, the Danish astronomer (the inventor of the 
transit-instrument, meridian-circle, and prime vertical instrument, — a man 
ahnost a century in advance of his day), found that the eclipses of Jupiter's 
satellites show a peculiar variation in their times of occurrence, which he 
explained as due to the time taken hy lic/JU to pass tlirougli space. His bold 
and original suggestion was neglected for more than fifty years, until long 
after his death, when Bradley's discovery of aberration proved the correct- 
ness of his views. 

441. Eclipses of the satellites recur at intervals which are really 
almost exactly equal (the perturbations being very slight), and 
the interval can easily be determined and the times tabulated. 
But if we thus predict the times of the eclipses during a whole 
synodic period of the planet, then, beginning at the time of oppo- 
sition, it is found that as the planet recedes from the earth the 
eclipses, as observed, fall constantly more and more behindhand, 
and by precisely the same amount for all four satellites. The dif- 
ference between the predicted and observed time continues to in- 
crease until the planet is near conjunction, when the eclipses are 
almost seventeen minutes later than the prediction. After the 
conjunction they quicken their pace and make up the loss, so that 
when opposition is reached once more they are again on time. 

It is easy to see from Fig. 149 that at opposition the planet 
is nearer the earth than at conjunction by just two astronomical 



units, i.e., JB — JA = 2 SA. Light coming from J to the earth 
when it is at A will, therefore, make the journey quicker than 
when it is at 7?, by twice the time it takes light to pass from S 
to A, provided it moves through space at a uniform rate, as 
there is every reason to believe. 

The whole apparent retardation of eclipses between opposi- 
tion and conjunction must, therefore, be exactly twice the time 
required for light to come from 
the SU71 to the earth. In this 
way the " light-equation con- 
stant " is found to be very 
nearly 499 seconds, or 8"^19% 
with a probable error of per- 
haps two seconds. 

Attention is specially di- 
rected to the point that the 
observations of the eclipses 
of Jupiter's satellites give 
directly neither the velocity 
of light nor the distance of the 
sun; they give only the time 
required by light to make 

the journey from the sun. Many elementary text-books, espe- 
cially the older ones, state the case carelessly. 

Since these eclipses are gradual phenomena, the determination of the 
exact moment of a satelhte's disappearance or reappearance is very 
difficult, and this renders the result somewhat uncertain. Prof. E. C. 
Pickering of Cambridge has proposed to utilize photometric observations 
for the purpose of making the determination more precise, and two series 
of observations of this sort and for this purpose are now completed, and 
are being reduced, one in Cambridge, and the other in Paris under the 
direction of Cornu, who devised a similar plan. Pickering has also applied 
photograpJiy to the observation of these eclipses with encouraging success. 

442. The Distance of the Sun determined by the << Light- 
Equation.'* — Until 1849 our only knowledge of the velocity of 

How this 
the constant 
of the light- 

The con- 
499^ ± 2\ 

Fig. 149. — Determination of the Equation 

of Light 

metliod of 
the eclipses. 



Distance of 
suu obtained 
by multiply- 
ing the 
constant of 
the light- 
equation by 
the velocity 
of light. 

light was obtained from such observations of Jupiter's satellites. 
By assuming as known the earth's distanee from the sun, the 
velocity of light can be obtained when we know the time occu- 
pied by light in coming from the sun. At present, however, 
the case is reversed. We can determine the velocity of light 
by two independent experimental methods, and with a surpris- 
ing degree of accuracy. Then, knowing this velocity and the 
"light-equation constant," tve can deduce the distance of the sun. 
According to the latest determinations, the velocity of light is 
186330 miles per second. Multiplying this by 499, we get 
92 979000 miles for the sun's distance. (Compare Sec. 173.) 

Saturn: its 
and varia- 
tions of 
its light. 



443. Saturn is the most remote of the planets known to the 
ancients. In brilliance it is inferior to Venus and Jupiter, or even 
Mars when nearest; still, it is a conspicuous object of the first 
magnitude, outshining all the stars (except Sirius) with a steady, 
yellowish radiance, not varying much in appearance from month 
to month, though in the course of fifteen years it alternately 
gains and loses nearly fifty per cent of its brightness with the 
changing phases of its rings ; for it is unique among the heavenly 
bodies, a great globe attended by eight, perhaps nine, satellites, 
and surrounded by a system of rings which has no counterpart 
elsewhere in the universe, so far as known at present. 

444. Orbit. — Its mean distance from the sun is about nine 
and one-half astronomical units, or 886 000000 miles ; but the 
distance varies nearly 100 000000, on account of the consider- 
able eccentricity of the orbit (0.056). Its nearest opposition 
approach to the earth is about 774 000000 miles, while at the 
remotest conjunction it is 1028 000000 miles away. 

The sidereal period of the planet is about twenty-nine and 
one-half years, the synodic being 378 days. The inclination of 
the orbit to the ecliptic is about 2^°. 


445. Dimensions, Mass, etc. — The apparent mean diameter of Diameter, 
the planet varies, accordhiff to the distance, from IV to 20'^ obiateness, 

-^ _ ^ _ ^ _ ^ surface, and 

The equatorial diameter is about 75000 miles, the polar diame- volume. 
ter only 67400; the mean diameter, therefore, is about 72500, 
— a little more than nine times the diameter of the earth. The 
ohlateness of Saturn (the flattening at the poles) is nearly J^, 
being greater than that of any other planet. 

The surface is about eighty-four times that of the earth, and 
its volume 768. 

Its mass is found by means of its satellites to be ninety-five Mass, 
times that of the earth, so that its mean density comes out only *^^^"^'^5'' '^"'^ 

'^ *^ surface 

one eighth that of the earth, — only two thirds that of water ! gravity. 
It is by far the least dense of all the planetary family. Its Saturn less 

^ . -, .., ^^. . , clense than 

mean superficial gravity is about 1.2 tmies gravity upon the ^vater. 
earth, varying, however, nearly twenty-five per cent between 
the equator and the pole. 

The rotation period is about 10^14™, as determined by Pro- Rotation 
fessor Hall in 1876 from a white spot that appeared on the ^^^J^^^l 
planet's surface and continued visible for several weeks. Later 
observations of Stanley Williams in 1893, while confirming this 
result, indicate that there are vigorous surface currents, as on 
Jupiter, so that different spots give rotation periods differing 
by several minutes. 

The equator of the planet is inclined about 27° to the plane its equator 
of the orbit. "'^^^^'"^ -^' 

to plane of 

446. Surface, Albedo, Spectrum. — The disk of the planet, like its orbit. 
that of Jupiter and the sun, is darker at the edge, and, like that 

of Jupiter, it show^s a number of belts arranged parallel to the 
equator. The equatorial belt is very much brighter than the 
rest of the surface (not quite so much so, however, as repre- 
sented in Fig. 150), and is often of a delicate pinkish tinge. 
The belts in higher latitudes are comparatively faint and 
narrow, while just at the pole there is a dark cap, sometimes dis- 
tinctly olive-green in color. Compared with Jupiter, however, 



there is very little detail observable on the surface; the 
edges of the belts are usually smooth, with only occasional 

Fig. 150. — Saturn 
After Proctor 

irregularities, and the spots, when they appear, are as a rule ill- 
defined and very faintly contrasted with the background, so 
that they are difficult to observe. Like the markings 


warm from 


Jupiter, they are almost certainly atmospheric, i.e.^ clouds of 
no great density. 

The mean albedo of the planet is 0.52, according to ZoUner, Albedo 0.52. 
— very nearly the same as that of Venus. 

The spectrum of Saturn is substantially like that of Jupiter, Spectrum of 
but the dark bands in the lower part of the spectrum are more *'^® planet. 

Bands more 

pronounced. These bands, which are doubtless due to some pronounced 
unidentified constituent of the planet's atmosphere, do not appear, ^^^^ ^^ ^^^^ 
however, in the spectrum of the rings, which presumably have 
very little atmosphere upon them. 

As to the physical condition and constitution of the planet, Probably 
it is probably essentially like that of Jupiter, though still farther 
from solidity ; it does not, however, seem to boil quite so vigor- sources, 
ously at the surface. Its supply of solar heat and light is less 
than -Jq- of that which we receive on the earth. 

447. The Rings. — The most remarkable peculiarity of the 
planet is its ring system. The globe is surrounded by three thin, its ring 
flat, concentric rings in the plane of Saturn's equator, like cir- ^y^^®^- 
cular disks of paper perforated through the center. They 
are generally referred to as A, B, and C, A being the exterior 

Galileo half discovered them in 1610; that is, he saw with Haifdis- 
liis little telescope two appendages on each side of the planet, ^-f^J^|*®^^ ^^' 
but he could make nothing of them, and after a Avhile he lost 
them, to regain them again some years later, greatly to his 

The problem remained unsolved for nearly fifty years, until Discovery 
Huyghens explained the mystery in 1655. Twenty years later coinpieted 
D. Cassini_ discovered that the ring is double, i.e., composed of giiens and 
two concentric rings, with a dark line of separation between ^- Passim, 
them, and in 1850, Bond of Cambridge, U.S., discovered the 
third "dusky" or "gauze" ring between the principal ring and Gauze ring 
the planet. (It was discovered a fortnig-ht later, and inde- discovered 

^ ° by Bond 

pendently, bv Dawes in England.) ini850 



of tlie rings. 

Rings ex- 

The outer ring, A, has an exterior diameter of about 173000 
miles and a width of not quite 12000. Cassini's division 
between this and B is about 1800 miles wide; the ring B, much 
the broadest and brightest of the three, has a breadth of about 
17000 miles. The semi-transparent ring, C, has a width of 
about 11000 miles, leaving a clear space of from 7000 to 
8000 miles in width between the planet's equator and its inner 
edge. Their thickness is exceedingly small, — probably less 
than 50 miles. (These dimensions are from the recent measure- 
ments of Professor See and differ slightly from those given in 

Fig. 151. — The Phases of Saturn's Rings 

once in 
years, when 
passes the 
nodes of 
its orbit. 

the G-eneral Astronomy.) There is some reason to suspect that 
the rings may have changed their dimensions at different times, 
but as yet the proof is insufficient. 

448. Phases of the Rings. — The rings are inclined about 28° 
to the ecliptic (27° to the planet's orbit), having their nodes in 
longitude 168° and 348°, and of course maintain their plane 
parallel to itself at all times. Twice in a revolution of the 
planet (once in about fifteen years) this plane sweeps across the 
orbit of the earth (too small to be shown in Fig. 151), occupy- 
ing not quite a year in so doing; and whenever the plane passes 
between the earth and the sun the dark side of the ring is 
towards us and the edge alone is visible. The plane of the 


ring traverses the orbit of the earth in about 359.6 days, and 
during this time the earth herself passes the plane eitlier once 
or three times, according to circumstances, — usually three times, 
thus causing two periods of disappearance during the critical 
year. When the earth is crossing the plane of the ring, so that 
its edge is exactly towards us, the ring becomes absolutely 
invisible to all existing telescopes for several days; and in the 
longer periods, while the dark side of the ring is presented to 
us, — sometimes for several weeks, — only the most powerful 
instruments can see it, like a fine needle of light piercing the 
planet's ball, and with satellites strung like beads upon it, as 
shown in the upper view of Fig. 150. 

449. The Structure of the Rings. — It is now universally structure of 
admitted that the rings are not continuous sheets, either solid or ^^eimgs. 

c" 'a swarm of 

liquid, but ?i flock or swarm of separate particles, little "moon- independent 
lets," each pursuing its own independent circular orbit around i^^ooniets. 
the planet, though all moving nearly in the same plane. 

The idea was first suggested by J. Cassini in 1715, and later by Wright 
in 1750, but was quite lost sight of until brought forward again by G. P. 
Bond in connection with his father's discovery of the dusky ring. Peirce Origm and 
soon demonstrated that the rings could not be solid, though he was dis- develop- 
posed to think they might be liquid. Clerk Maxwell in 1857 went further, ^^^^^ ^^ *^® 
by showing mathematically that while they could be neither solid nor 
liquid, they must, in order to be permanent, be constituted as explained 

There are also observational facts that confirm the theory. Photometric 
Seeligfer has shown that the variations in the naked-eve bright- ^^s^^^^- 

^ . . tions of 

ness of the planet, due to the phases of the rings, can be explained seeiiger and 
only on the hypothesis that they are like clouds of dust. Again, Barnard, 
in 1892 Barnard observed the satellite lapetus during one of 
its eclipses (a very rare event) and found that the shadow of 
the ring is not opaque; the satellite did not disappear when 
immersed in it, but vanished as soon as it entered the shadow 
of the ball. 



scopic proof 
that the 
outer edge 
of the ring 
moves more 
slowly than 
the inner. 

Effect of 
reflection on 
shift of 

1 Millimeter 

50 km. 

L- 50 km. 

450. Keeler's Demonstration of the Meteoric Theory of Saturn's 
Rings. — In 1895 Keeler, then at Allegheny, obtained spectro- 
scopic proof that the outer edge of the ring revolves more slowly 
than the inner, as the theory requires, but as would not be the 
case if the ring were a continuous sheet. Photographs were 
made of the spectrum of the planet with the slit of the spectro- 
scope crossing the planet 
and its rings, as shown 
in Fig. 152, which is a 
much magnified draw- 
ing of the actual image. 
At the western limb 
of the planet and the 
western extremity of the 
ring the motion of rota- 
tion carries the particles 
from us, and the dis- 
placement of the spec- 
trum lines should be 
towards the red, accord- 
ing to Doppler's prin- 
ciple ; moreover, since 
the particles shine by 
reflected sunlight, the 
displacement is practically doubled at the time of the planet's 
opposition, — tivice as great as if the particles were self-luminous. 
On the eastern side there is an equal shift towards the violet. 
Now, on looking at the diagram of the spectrum (given below 
the planet), we see that while at C the line in the spectrum 
is bodily displaced towards the red, as it ought to be, the 
displacement at the outer edge of the ring is less than that at 
the inner, and correspondingly at A. This shows that the 
particles at the outer edge are moving more slowly than at the 

Fig. 152. — Spectroscopic Observation of 

Saturn's Ring 



The fact is made conspicuous by its effect upon the inclina- 
tion of the lines: while in the spectrum of the ball the lines 
slope upwards towards the right, in the ring spectrum on both 
sides they slope the other way. 

At the inner edge of the ring the observations indicated a 
velocity of 12i miles a second, at the outer edge only 10, — 
precisely the velocities that satellites of Saturn ought to have 
at the corresponding distances from the planet. 

It may be noted also that the inclination of the lines on the ball indi- 
cates at the edge of the planet a velocity of 6.4 miles a second, correspond- 
ing to a rotation period of 10^14*". 6, — almost exactly agreeing with that 
deduced by Professor Hall from the observation of the spot. 

The observations are extremely delicate, as the whole width of the 
spectrum was not quite a millimeter, the figure being magnified nearly 
fifty times. But Keeler's results have since been fully confirmed by 
Deslandres, Belopolsky, and Campbell. 

The investigations of Hermann Struve show that the mass Mass of 
of the rings is inappreciable ; they produce no distinctly "^^^ 

m PP 1 . p 1 ^^^ rr. 1- insensible. 

observable ertect upon the motion oi the satellites, io use his 
graphic expression, they seem to be composed of " immaterial 
light," — mere dust films or wreaths of fog. 

451. Stability of the Ring. — If the ring were solid, it cer- Question of 
tainly would not be stable ; it could not endure the strains ^tabihty of 

1 . . . . . ^ ,.T the rings. 

due to its rotation, nor is it certain that even the swarmlike 
structure makes it forever secure. There have been strong 
suspicions of a change in the width of the rings and their 
divisions, but the latest measurements hardly confirm the idea. 
It is not, however, improbable that the ring may ultimately be 
broken up. 

It can hardly be doubted that the details of the ring are con- 
tinually changing to some extent; thus, the outer ring, A^ is 
occasionally divided into two by a very narrow black line known 
as " Encke's division," though more usually there is merely a 
darkish streak upon it not amounting to a real break in the 



of satellites. 


surface. When the rings are edgewise notable irregularities 
are observed upon them, as if they were not accurately plane 
nor quite of even thickness throughout. Irregularities are 
reported also in the form of the shadow cast by the planet on 
the rings, indicating that the ring surface is not entirely flat. 

But caution must be used in accepting and interpreting such observa- 
tions, because illusions are very apt to occur from the least indistinctness 
of vision or faintness of light. Generally speaking, the writer has found 
that the better the seeing, the fewer abnormal appearances are noted, and 
the experience of other observers with large telescopes is the same. 

452. Satellites. — Saturn has eight (possibly nine) of these 
attendants, the largest of which, named Titan, was discovered 
by Huyghens in 1655. It is easily seen with a 3-inch telescope. 

D. Cassini, with his long-focus telescope (Sec. 44, note), found 
four others before 1700; Sir W. Herschel in 1789, with his 
great 4-foot reflector, discovered the two which are nearest the 
planet; and in 1848 the elder Bond added an eighth. 

As the order of discovery does not agree with that of distance, 
it has been found convenient, in order to avoid confusion, to 
adopt names for the satellites (suggested by Sir John Herschel). 
They are, beginning with the most remote, 

lapetus, (Hyperion), Titan ; Rhea, Dione, Tethys ; Enceladus, Mimas. 

Leaving out Hyperion (which had not been discovered when 
the names were first assigned), they form a line and a half of a 
regular Latin pentameter. 

The range of the system is enormous. lapetus is at a distance 

of 2 225000 miles, with a period of seventy-nine days, — nearly 

Peculiarities as long as that of Mercury. On the western side of Saturn this 

of lapetus. satellite is always much brighter than at the eastern, showing that, 

like our own moon, it always keeps the same face towards the 

planet, — one half of its surface being darker than the other. 

Titan. Titan, as its name suggests, is by far the largest. Its distance 

is about 770000 miles and its period a little less than sixteen 



days. It is probably 3000 or 4000 miles in diameter. Its mass 
is found, from the perturbations produced by it in the motion 
of the other satellites, to be ^gVo" ^^ Saturn's. 

The orbit of lapetus is inclined about 10° to the plane of the 
rings, but all of the other satellites move sensibly in their plane, 
and all the five inner ones in orbits sensibly circular. 

Early in 1899 Prof. W. H. Pickering announced the discovery of a 
ninth satelHte (to which he has assigned the name of Phcebe^, found on 
photographs made at Arequipa, the southern annex of the Harvard College Phoebe, a 
Observatory. The data were not sufficient to determine its orbit further probable 
than that the satellite must be at a distance of several millions of miles, — 
much exceeding that of lapetus, and with a period much exceeding a year. 

'No confirmation has, however, yet been obtained from photographs 
since made. The satellite, if it exists, is too small to be seen with any 
existing telescope ; but we can photograph what is not visible. 


453. Discovery of Uranus. — Uranus was tlie first planet ever Discovery 
" discovered," and the discovery created great excitement and 
brought the highest honors to the astronomer. It was found in I78i 
accidentally by the elder Herschel on March 13, 1781, while 
"sweeping" the heavens for interesting objects with a 7-inch 
reflector of his own construction. He recognized it at once by 
its disk as something different from a star, but never dreaming 
of a new planet supposed it to be a peculiar kind of comet ; its 
planetary character was not demonstrated until nearly a year 
had passed, when Lexell of St. Petersburg showed by his calcu- 
lations that it was doubtless a planet beyond Saturn, moving in 
a nearly circular orbit. 

It is easily visible to a good eye on a moonless night as a 
star of the sixth magnitude. 

The name of Uranus, suggested by Bode, finally prevailed 
over other appellations that were proposed (Herschel had called 
it the " Georgium Sidus," in honor of the king). 

of Uranus 
by Herschel 



tions of the 

Data relat- 
ing to its 

and bulk of 
the planet. 

Its mass and 

Albedo and 

It was found on reckoning backward that the planet had been 
many times observed as a star and had barely missed discovery 
on several previous occasions. Twelve observations of it had 
been made by Lemonnier alone, and later they proved extremely 
valuable in connection with the investigations which led to the 
discovery of Neptune. 

454. Orbit of Uranus. — The mean distance of the planet from 
the sun is 19.2 astronomical units, or 1782 000000 miles. The 
sidereal period is eighty-four years and the synodic 369-g- days, 
the annual advance of the planet among the stars being only 
a little over 4^°. The eccentricity of the orbit is about the 
same as that of Jupiter, the sun being 83 000000 out of the 
center of the orbit. The inclination of the orbit to the ecliptic 
is only 46'. The light and heat received from the sun are only 
about -g^Q- of that received by us. 

455. The Planet itself. — In the telescope it shows a greenish 
disk about 4'' in diameter, though the measurements of See 
make it only 3''. 3, corresponding to a diameter of only 28500 
miles, which is 3400 miles less than that hitherto generally 
accepted and given in the tables of the Appendix. If we admit 
the correctness of this new measure, the volume comes out only 
forty-seven times that of the earth as against the sixty-five of 
the tables. The mass is determined much more accurately than 
the diameter (by the motion of its satellites) and is about 14.6 
times that of the earth, the density of the planet (still accepting 
See's diameter) being 0.31 of the earth and its surface gravity 
a little greater than ours, 1.11. The albedo of the planet is 
very high, 0.62 according to Zollner, even higher than that of 
Jupiter. The spectrum exhibits strong dark bands in the red, 
due, doubtless, to some unidentified substance in the planet's 
probably dense atmosphere. They explain the greenish tinge 
of the planet's light. 

The planet's disk as determined by various observers about 
1882, when the plane of the satellites' orbits was directed 


towards the earth, was obviously oval, indicating an oblateness Obiateness. 
of about jL. At present (1902) the pole is presented to us and 
the disk appears round. There are no distinct markings on the 
disk, but there are faint traces of belts, Avhich appear to lie not Belts, 
exactly in the plane of the satellites, but at an angle of some 
15° or 20°. They are too indistinct, however, to warrant any 
positive assertion. Nothing has yet been observed from which 
the rotation of the planet can be determined. 

456. Satellites. — The planet has four satellites, — Ariel, Um- The four 
briel, Titania, and Oberon, Ariel being the nearest to the planet. ^^ ® 

The two brightest, Oberon and Titania, were discovered by 
Sir William Herschel, who thought he had discovered four 
others also;- he may have glimpsed Ariel and Umbriel, but it 
is very doubtful. They were first certainly discovered and 
observed by Lassell in 1851. 

These satellites, especially the two inner ones, are telescopi- 
cally the smallest bodies in the solar system and the hardest to 
see, excepting perhaps the fifth satellite of Jupiter. In real 
size they are, of course, much larger than the satellites of Mars, 
very likely measuring from 200 to 500 miles in diameter. 

Their orbits are sensibly circular, and all lie in one plane. Their orbits. 
which ought to be, and probably is, coincident with the plane 
of the planet's equator ; but the belts raise questions. They 
are very " close packed " also, Oberon having a distance of only 
375000 miles and a period of 13^^11^\ while Ariel has a period 
of 2'^11^^ at a distance of 120000 miles. Titania, the largest 
and brightest, is at a distance of 280000 miles, a little greater 
than that of the moon from the earth, with a period of 8*^^17^. 

The most remarkable point about this system remains to be 
mentioned. The plane of their orbits is inclined 82°. 2 to the Great incii- 
plane of the ecliptic, and in that plane they revolve hackivards; ^^^^onof 

^ ^ ^ -^^ . _ the orbits. 

or we may say, what comes to the same thing, that their orbits Backward 
are inclined to the ecliptic at an angle of 97°. 8, in which case revolution 

of satellites 

their revolution is to be considered as direct. 



of Uranus. 

of the dis- 
theory and 


457. Discovery of Neptune This is reckoned as the greatest 

triumph of mathematical astronomy since the days of Newton. 
Intractability It was very soon found impossible to reconcile the old observa- 
tions of Uranus by Lemonnier and others with any orbit that 
would fit the observations made in the early part of the nine- 
teenth century, and, what was worse, the planet almost imme- 
diately began to deviate from the orbit computed from the new 
observations, even -after allowing for the disturbances due to 
Saturn and Jupiter. It was misguided by some unknown influ- 
ence to an amount almost perceptible by the naked eye; the 
difference between the actual and computed places of the planet 
amounted in 1845 to the "intolerable quantity" of nearly two 
minutes of arc. 

This is a Httle more than one half the distance between the two prin- 
cipal components of the doable-double star, e Lyrse, the northern one of 
the two little stars which form the small equilateral triangle w ith Vega 
(Fig. 190, Sec. 585). A very sharp eye can perceive the duplicity of e 
without the aid of a telescope. 

One might think that such a minute discrepancy between 
observation and theory was hardly Avorth minding, and that to 
consider it '' intolerable " was putting the case very strongly, 

De minimis but in science unexplained " residuals " are often the seeds 
from which new knowledge springs. Just these minute dis- 
crepancies supplied the data which sufficed to determine the 

Mathemati- position of a great world, before unknown. 

cai discovery ^^ ^^^ result of a most skilful and laborious investigation, 

of Neptune ^ _ ^ 

byLeverrier. Leverrier, a young French astronomer, wrote in substance to 
Galle, then an assistant in the Observatory at Berlin: 

'-'•Direct your telescope to a point on the ecliptic in the constella- 
tion of Aquarius^ in longitude 326°^ and you will find within a 
degree of that place a neiv planet^ looMng like a star of about the 
ninth magnitude^ and having a perceptible dishy 



The planet was found at Berlin on the night of Sept. 2-3, Optical dis- 
1846, in exact accordance with this prediction, within half an ^overyby 

. . . , Galle, 1840. 

hour after the astronomers began looking for it and within 52' 
of the precise point that Leverrier had indicated. 

The English Adams fairly divides with Leverrier the honors for the Share of 

mathematical discovery of the planet, having solved the problem and Adams hi 

dednced the planet's approximate place even earlier than his competitor. ® ^^" 

The planet was being searched for in England at the time when it was 

found in Germany. It had, in fact, been already twice observed, and the 
discovery would necessarily have followed in a few weeks, upon the reduc- 
tion of the English observations. Leverrier died in 1877, Adams in 1892. 
Galle alone of the discoverers survives, the aged Emeritus Professor and 
Director of the University Observatory at Breslau. 

458. Error of the Computed Orbit. — 3oth Adams and Lever- Error of 
rier, besides calculating the planet's position in the heavens, ^^JJ^f^d^ 
had deduced elements of its orbit and a value for its mass, to the 
which turned out to be seriously incorrect. The reason was assumption 

-^ . ofBode's 

that they assumed that the new planet's mean distance from the Law, which 
sun would follow Bode's Law, a supposition quite warranted by ^®^*® breaks 
all the facts then known, but which, nevertheless, is not even 
roughly true. As a consequence, their computed elements were 
erroneous, and that to an extent which has led high authorities 
to declare that the mathematically computed planet was not 
Neptune at all, and that the discovery of Neptune itself was 
simply a "happy accident." 

This is not so, however. While the data and methods 
employed were not by themselves sufficient to determine the 
planet's orbit with accuracy, they were adequate to ascertain Method 
the planet's direction from the earth; the computers informed "f® ^^^^^ 

^ ^ planet s 

the observers where to point their telescopes, and this was all direction 
that was necessary for finding the planet. In a similar case the ^^'^^^ earth. 
same thing could be done again. 

459. The Planet and its Orbit. — The planet's mean distance 
from the sun is a little more than 2800 000000 miles (instead of 



Data relat- 
ing to 

of Neptune. 

Diameter of 
the planet 
smaller than 

Mass and 

Albedo and 

being over 3600 000000, as it should be according to Bode's 
Law). The orbit is very nearly circular, its eccentricity being 
only 0.009. Even this, however, makes a variation of over 
50 000000 miles in the planet's distance from the sun. The 
period of the planet is about 165 years (instead of 217, as it 
should be according to Leverrier's computed orbit) and the 
orbital velocity is about 3^ miles per second. The inclination 
of the orbit is about If °. 

Neptune appears in the telescope as a small star of between 
the eighth and ninth magnitudes, absolutely invisible to the 
naked eye, but easily seen with a good opera-glass, though not 
distinguishable from a star with a small instrument. Like 
Uranus, it shows a greenish disk, having an apparent diameter, 
according to the measures of H. Struve, of 2". 2. The measures 
of earlier observers were all much larger, and until very recently 
the value 2''. 6 was generally accepted, and is for the present 
allowed to stand in the Appendix tables. Recent measures of 
Struve, Barnard, and See all concur, however, in showing that 
this value is much too large. 

Accepting Struve's measures, the diameter comes out only 
29750 miles, and its volume fifty-three times that of the earth; but 
the margin of possible error must be still quite large. The mass^ 
as determined from its satellite, is about seventeen times that of 
the earth. Its density (according to Struve's diameter) comes out 
0.34, and the surface gravity one and one-fourth times our own. 

The planet's albedo, according to Zollner, is 0.46, — a trifle 
less than that of Saturn and Venus. 

There are no visible markings upon its surface, and nothing 
certain is known as to its rotation. 

The spectrum of the planet appears to be like that of Uranus, 
but of course is rather faint. 

It will be noticed that Uranus and Neptune form a '' pair of 
twins," very much as the earth and Venus do, being almost alike 
in magnitude, density, and many other characteristics. 


460. Satellite. ~ Neptune has one satellite, discovered by Neptune's 
Lassell within a month after the discovery of the planet ^^^^ ^^^' 
itself. Its distance is about 223000 miles and its period 5'^21^\ 

Its orbit is inclined to the ecliptic at an angle of 34° 48' and it 
moves backward in it from east to west, like the satellites of 
Urajius. It is a very small object, not quite as bright as Oberon, 
the outer satellite of Uranus. From its brightness, as compared 
with that of Neptune itself, its diameter is estimated as about 
the same as that of our own moon. 

461. The Sun as seen from Neptune. — At Neptune's distance 
the sun has an apparent diameter of only a little more than one 
minute of arc, — about the diameter of Venus when nearest us, 
and too small to be seen as a disk by the naked eye, if there be 

eyes on Neptune. The light and heat there are only -gi ^ part Sunlight on 
of what we get at the earth. Still, we must not imagine that ^^ptune. 
the Neptunian sunlight is feeble as compared with starlight, 
or even with moonlight. At the distance of Neptune the sun 
gives a light nearly equal to 700 full moons, — about eighty 
times the light of a standard candle at one meter's distance, — 
and is abundant for all visual purposes. In fact, as seen from 
Neptune, the sun would look very like a 1200 candle-power 
electric arc at a distance of only 12 or 13 feet. 

462. Ultra-Neptunian Planets. — Perhaps the breaking down Possible 
of Bode's Law at Neptune may be regarded as an indication p^^"®^^ 
that the solar system ends there, and that there is no remoter Neptune. 
planet ; but of course it does not make it certain. If such a 
planet of any magnitude exists, it is sure to be found sooner or 

later, either by means of the disturbances it produces in the 
motion of Uranus and Neptune, or else by the methods of the 
asteroid hunters, — although its slow motion will render its 
discovery in that way difficult. Quite possibly it may come 
in a few years as the result of the photographic star-charting 
operations now in progress. 



1. When Jupiter is visible in the evening do the shadows of his satel- 
lites precede or follow the satellites as they cross the i)lanet's disk ? 

2. On which limb, the eastern or the western, do the satellites appear 
to enter upon the disk ? 

3. What probable effect would the great mass of Jupiter have upon 
the size of animals inhabiting it, if there were any ? 

4. How would sunlight upon Saturn compare with sunlight on the 
earth ? How with moonlight ? 

5. What would be the greatest elongation of the earth from the sun as 
seen from Jupiter ? from Saturn ? from Uranus ? 

6. What would be the apparent angular diameter of the earth when 
" transiting " the sun as seen from Jupiter ? 

7. What is the rate in miles per hour at which a white spot on the 
equator of Jupiter, showing a rotation period of 9^50"^, would pass a dark 
spot indicating a period of 9^55'"? 

8. Find the diameter, volume, density, and surface gravity of Neptune, 
accepting See's measured diameter of the planet, viz., 2'''.01, taking the 
planet's mass as 17 times that of the earth, the solar parallax as 8".80, 
and the mean distance of Neptune from the sun as 30.055. 

r Diameter, 27200 miles. 
Ans. -l Volume, 41 times the earth. 
[Density, 0.42. 



Importance and Difficulty of the Problem — Historical — Classification of Methods 
— Geometrical Methods — Oppositions of Mars and Certain Asteroids, and Tran- 
sits of Venus — Gravitational Methods 

463. In some respects the problem of the sun's distance is the 
most fundamental of all that are encountered by the astron- 
omer. It is true that many important astronomical facts can 
be ascertained before it is solved : for instance, by a method 
which has been given in Sec. 371, we can determine the rela- Relative 
tive distances of the planets and form a map of the solar system, ^^^^tances m 

^ ^ ^ -J ' solar system 

correct in all its proportions^ although the unit of measurement easily deter- 
is still undetermined, — a map ivithout any scale of miles. But """^d- 
to give the map its use and meaning, we must ascertain the 
scale, and until we do this we can have no true conception of 
the real dimensions, masses, and distances of the lieavenly 
bodies as compared with our terrestrial units of mass and dis- 
tance. Any error in the assumed value of the astronomical 
unit propagates itself proportionally through the whole system, 
not only solar but stellar. 

The difficulty of the problem equals its importance. It is no importance 
easy matter, confined as we are to our little earth, to reach out ^^^^^ *^^^" 

-^ ' culty of 

into space and stretch a tape-line to the sun. In Sees. 173 and determining 
442 we have already given the two -methods of determining the absolute 
sun's distance, which depend on our experimental knowledge 
of the velocity of light. They are satisfactory and sufficient 
for the purposes of the text. But methods of this kind have 
become available only since 1849. 




Solar par- 
allax 3', 
according to 

Proof that 
this must he 
too great 
hy Kepler 
aud Cassini. 

from transits 
of Venus in 
1761 and 

tion of 

Previously astronomers were confined entirely to purely 
astronomical methods, depending either upon geometrical meas- 
urement of the distance of one of the nearer planets when 
favorably situated, or else upon certain gravitational relations 
which connect the distance of the sun with some of the irregu- 
lar motions of the moon, or with the earth's power of disturbing 
her neighboring planets, Venus and Mars. 

464. Historical. — Until nearly 1700 no even approximately 
accurate knowledge of the sun's distance had been obtained. 
Up to the time of Tycho it was assumed on the authority of 
Ptolemy, who rested on the authority of Hipparchus, who in 
his turn depended upon an observation of Aristarchus (erroneous, 
though ingenious in its conception), that the sun's horizontal 
parallax is 3', — a value more than twenty times too great. 

Kepler, from Tycho's observations of Mars, satisfied himself 
that the parallax certainly could not exceed 1', and was prob- 
ably much smaller; and at last, about 1670, Cassini also, by 
means of observations of Mars made simultaneously in France 
and South America, showed that the solar parallax could not 
exceed 10". 

The transits of Venus in 1761 and 1769 furnished data that 
proved it to lie between 8'' and 9'', and the discussion of all the 
available observations, published by Encke about 1824, gave as 
a result 8''. 5776, corresponding to a distance of about 95 000000 
miles. The accuracy of this determination was, however, by no 
means commensurate with the length of the decimal, and its 
error began to be obvious about 1860. It is now practically 
settled that the true value lies somewhere between 8 ''.75 and 
8".85, the sun's mean distance being between 92 400000 and 
93 500000 miles. Indeed, it is now certain that the figure 8''. 8, 
adopted in the text, must be extremely near the truth. 

The methods available for determining the distance of the 
sun may be classified under three heads, — geometrical, gravita- 
tional, and physical. The physical methods (by means of the 


velocity of light) have been already discussed (Sees. 173 and 
442). We proceed to present briefly the principal methods that 
belong to the two other classes. 


465. The direct geometrical method of determining the snn's Direct 
distance and parallax (by observing the sun itself at stations geo^f^etncai 

i^ \ -J ^ ^ _ method 

widely separated on the earth, in the same way that the dis- useless. 
tance of the moon is found — Sec. 196) is practically worthless. 
The parallax of the sun being only 8 ".8, the inevitable errors of 
the best direct observation would be far too large a fraction 
of the quantity sought. Moreover, the sun, on account of the 
effect of its heat upon an instrument, is a very unsatisfactory 
object to observe. 

Since, however, we can compute at any time the distance of any 
planet from the earth ifi astronomical units, it will answer every 
purpose to measure the distance m miles to any one of them. 

466. Observations of Mars. — When Mars comes nearest to indirect 

the earth its distance from us can be measured with reasonable determina- 
tion by 
accuracy in either of two ways : observa- 

(1) By observations from two or more stations widely separated ^^^"^ ^^ 

. ■ Mars; two 

tn taCtC ttvve* inptlinrl^ 

(2) By observations of the planet from a single station near 
the equator when the planet is near its rising and setting. 

In the first case the observations may be (a) meridian-circle 
observations of the planet's zenith-distance, exactly such as 
are used for getting the distance of the moon (Sec. 196); or 
(b) they may be micrometer or heliometer measures of the differ- First 
ence of declination between the planet and stars near it; or, "method 


linally, (c) instead of measuring this distance directly photo- able in 

graphs may be taken and measured later. requiring 

Since, however, at least two different observers and two servers and 

different instruments are concerned in the observations, the instruments. 



method : 
a single 
taking ad- 
vantage of 
the earth's 

of planet's 
from neigh- 
boring stars. 

results are less trustworthy than those obtained by the second 

467. In this case a single observer, by measuring (best with a 

heliometer — Sec. 72) the apparent distance between the planet 

and small stars nearly east and west of it at the times when the 

planet is near the horizon, can determine its par- 

allax with great accuracy. 

Fig. 153 illustrates the principle involved. 
When the observer at A (a point on or near the 
earth's equator) sees the planet M just rising, he 
sees it at a (a point on the celestial sphere east 
of (?, the point where it would be seen from 
the center of the earth), the angle C3fA being 
the planet's equatorial horizontal parallax. 

Twelve hours later, when the rotation of the 
earth has taken the observer to B and the planet 
is setting, he sees it at 5, displaced by parallax 
just as much as before, but to the west of c, its 
geocentric place. In other words, when the 
planet is rising the parallax increases its right 
ascension, and when setting diminishes it. 

Suppose, now, that for the moment the orbital 
motion of the planet and the earth are suspended, 
the planet being at opposition and as near the 
earth as possible. If, then, when the planet is 
rising we measure the apparent distance, MgS 
(Fig. 154), from a star, S^ and twelve hours later measure it 
again, the distance 31^ to 31,^ will be twice the horizontal paral- 
lax of the planet. The earth's rotation will have carried the 
observer a long journey, transporting him and his instrument, 
without disturbance, expense, or trouble, to a (virtually) different 
station 8000 miles away. 

In practice the observations, of course, cannot be made at the 
moment when the planet is exactly on the horizon, but they are 

Fig. 153 



kept up during the whole time while the planet is crossing the 
heavens. Moreover, measures are made not from one star 
only, but from all that are in the planet's neighborhood. The 
orbital motion, both of the j)lanet and of the earth, during the 
observations must also be allowed for; but this presents no 
serious difficulty. 

The most important application of this method was in 1877, 
at Ascension Island, by Gill (now Sir David Gill of the Cape 
of Good Hope Observatory). He got for the solar parallax 
8".783 ± .015. The size of the planet's disk, its brightness, 
and its "phase" (except at the 
moment of opposition), however, 
interfere somewhat with the pre- 
cision of the necessary measure- 
ments, and the great difference 
of brightness between it and the 
stars makes it difficult to use 

468. Observations of Asteroids. 
— The asteroids do not present 
the same difficulties in determin- 
ing their apparent places among 
the stars by heliometer measures 
or photography, since they them- 
selves are mere starlike points. 
Although none of them (except 

Eros) come quite as near to the earth as Mars, several of them 
come near enough to make it possible to obtain from their 
observations results notably more satisfactory than those from 

The heliometer observations of Iris, Sappho, and Victoria, 
made by Gill at the Cape of Good Hope in 1889-91, in concert 
with several other heliometer observers in Europe and America, 
gave 8^'. 802 ± .005. In this case, of course, the method used 













Fig. 154. — Micrometric Coinparisou 
of Mars with Neigbboriug Stars 

tions of 
Gill at 

by observa- 
tions of 




observed in 

Transits of 
Venus: her 
ment on 
sun's disk is 
2.61 times 
her parallax. 

principle by 

which solar 

was neither (1) nor (2) of Sec. 466 exclusively. The apparent 
displacements of the planets, clue both to the distance between 
the stations and to the motion of the stations on the whirling 
earth, all contribute to the result, complicating the calculation, 
but increasing its precision. 

We have already spoken of the case of Eros (Sec. 427). The 
observations of 1900-01, largely photographic, ought ^ when they 
have been thoroughly discussed, to give a result even more pre- 
cise than that last quoted. In 1931, if the weather is good for 
a few days at the critical time when the planet is nearest, the 
opportunity will be still more favorable. 

469. Transits of Venus. — When Venus is at or near inferior 
conjunction her distance is less than that of Mars at opposi- 
tion; but she cannot be observed for parallax in the same way, 
because she is then in the twilight, and little stars near her 
cannot be seen for use as reference points. Now and then, 
however, she passes between us and the sun and " transits " 
the disk, as explained in Sec. 405. Her distance from the 

earth is then only 26 000000 miles, 
and her parallax is much greater 
than that of the sun. Seen by two 
observers at different stations on the 
earth, she will therefore appear to be 
projected on two different points of 
the sun's disk, and her apparent angu- 
lar displacement on the sun's surface 
will be the difference between her 
own parallactic displacement (corre- 
sponding to the distance between the 
two stations) and that of the sun itself. This relative displace- 
ment is Jl^f, or 2.61, times as great as the displacement of 
the sun. 

To determine the solar parallax, then, by means of the tran- 
sit of Venus, we must somehow measure the apparent distance 

Fig. 155. 

Contacts in a Transit 
of Venus 


in seconds of arc between two positions of Venus on the sun's parallax is 

disk, as seen simultaneously from two widely distant stations f^^termmed 

, '^ from transit. 

of known latitude and longitude on the earth's surface. 

The methods earliest proposed and executed depend upon Methods 

observations of the instant of contact between the planet and ^^pe^^^^^s 

■^ on oDserva- 

the sun's limb. There are four of these contacts, as shown in tions of the 

Fig. 155, the first and fourth external^ the second and third contacts. 

470. Halley's Method, or the Method of Durations. — Halley Haiiey's 

was the first to notice, in 1679, the peculiar advantas^es of the ^^®*^^o^- 

*» 1 • • 1 1 • c f^^^i'ations 

transit of Venus as a means for determining the distance of observed 

the sun, and he proposed a method which consists in simply ^^"^"^ 

observing the duration of the transit at stations chosen as far f^r apart in 

apart in latitude as possible. latitude. 

This had the great advantage of not requiring an accurate Advantages 

knowledge of the longitudes of the stations, which in his time ^°^ disad- 
would have been very difficult to determine, nor did it require 






Fig. 15(5. — Haiiey's Method 

any knowledge of the absolute, or Greenwich, time of contact. 
It is only necessary to know the latitudes of the observers 
and that their timepieces should run accurately during the short 
time (five or six hours) while the planet is crossing the sun's disk. 
On the other hand, the stations must be in high latitudes, and 
the observer must see both beginning and end of the transit ; if 
he loses either on account of clouds, the method fails. 

Fig. 156 illustrates in a general way the principle involved: 
the two observers at JE and B see the planet crossing the sun's 
disk on the chords df and ac, respectively, and from the duration 



tions should 
give length 
of transit 
chords with 

not realized. 

Effect of 
the planet's 

and known rate of motion of the planet the length of the two 
chords in seconds of arc can be computed with more accuracy 
than it can be determined by any micrometer measure, provided 
the instant of contact can be accurately observed. But, since the 
angular semidiameter of the sun is known, the distances hS and 
eS oi the two chords from the center of the sun can be com- 
puted, and their difference, eh (all in seconds of arc), and that 
with very great accuracy if the chords fall, as they have done in 
all the transits yet observed, near the edge of the sun's disk. 

But, since VE and Ve are in the proportion of 277 to 723, 
eh (in miles) is J|^|- of EB^ provided the two stations are so 
chosen that the line EB is perpendicular to the plane of the 
planet's orbit (if not, due allowance must, and can, be made to 
get the "effective length" of EB). 

We have, then, eh, both in seconds and in miles; we know, 
therefore, how many miles go to one second of arc at the sun's 
distance. It comes out about 450, and therefore (Sec. 10, eh 
taking the place of r) the sun's distance is about 450 miles X 
206265, or 92 800000 miles. 

For details of the methods of accurate calculation from the 
actual observations, the reader must be referred to works deal- 
ing with the special subject. 

Halley expected to depend mainly on the two internal con- 
tacts, which he supposed could be observed with an error not 
exceeding a single second of time. If so, the observations 
would determine the sun's parallax within -^^-^ of its true value. 

Unfortunately, this accuracy is not found practicable. There 
are usually large errors caused by the imperfection of the tele- 
scope and eye of the observer, as well as atmospheric condi- 
tions. And even if these are avoided, the atmosphere of the 
planet introduces a difficulty that cannot be evaded; it pro- 
duces a luminous ring around the edge of the planet (Sec. 401, 
Fig. 137), which prevents any certainty as to the precise moment 
when the planet's disk is tangent to the limb of the sun. The 


contact observations during the last two transits in 18T4 and 
1882 were uncertain, under the very best conditions, by at least 
five or six seconds. 

471. Delisle's Method. — Halley's method requires stations in Deiisie's 

higfh latitudes, uncomfortable and hard to reach, and so chosen ^^®^^° • 

° , , observa- 

that both the beginning and end can be seen. And both must tions of 

be seen or the method fails. absolute 

1111 . f • instants of 

Delisle s method, on the other hand, employs pairs of stations contact from 

near the equator^ but as nearly as possible on opposite sides of equatorial 

the earth, and it does not require that the observer should see ^i^eiy 

both the beginning and end of the transit, — observations of separated, 
either phase can be utilized for their full value, which is a great 

Fig. 157. —Delisle's M&thod 

advantage, — but it requires that the loyigitudes of the stations 
should be known with extreme precision, since the method con- 
sists essentially in observing the absolute time of contact (^.e., 
Greenwich or Paris time at both stations). It is beautifully 
simple and easy to understand. 

An observer at E on the equator (Fig. 157) on one side of 
the earth notes the moment of internal contact in Greenwich 
time, the planet being then at V^ ; when IF, on the other side of 
the earth, notes the contact (also in Greenwich time) the planet 
will be at V^-, and the angle 1\DV^ is the earth's apparent 
diameter as seen from the sun, i.e., tivice the suris horizontal Theoretical 
parallax (Sec. 78). Now the angle at D is at once determined simplicity of 

. , . . . the method. 

by the time occupied by Venus in moving from J\ to V^. It is 
simply just the same fraction of 360° that this elapsed time is of 
58^ days^ the planet's synodic period. If, for example, the 



from lielio 
metric and 


difference of time were eleven and a half minutes between the 
contact as observed at E and W^ we should find the angle at D 
to be about 17 ".7. 

From all the contact observations^ several hundred in number, 
made during the transits of 1874 and 1882, Newcomb gets a 
solar parallax of 8''.794 ± .018. 

472. Heliometric and Photographic Observations. ^Instead of 
observing merely the four contacts and leaving the rest of the 
photometric transit unutilized, we may either keep up a continued series of 
measurements of the planet's position upon the sun's disk with 
a heliometer, or we may take a series of photographs to be 
measured up at leisure. Such heliometer measures or photo- 
graphs, taken in connection with the recorded Greenwich times 
at which they were made, furnish the means of determining 
just where the planet appeared to be on the sun's disk at any 
given moment, as seen from the observer's station. A compari- 
son of these positions with those simultaneously occupied by 
the planet, as seen from another station, gives at once the means 
of deducing the parallax. 

In 1874 and 1882 several hundred heliometer measures were 
made (mostly by German parties), and about six thousand 
photographs were obtained at stations in all quarters of the 
earth where the transits could be seen, — more than two thou- 
sand by the different American parties. The final result of 
all tliese observations is given by Newcomb as 8''. 857 ± .023, — 
differing to an unexpected degree from the figures given by 
other methods, and seriously discordant among themselves, as 
shown by the large probable error. 

tory result 

It looks as if measurements of this sort must be vitiated by some con- 
stant source of error as yet undetected. 

On the whole, the outcome of the two transits has been to satisfy 
astronomers that other methods of determining the sun's parallax are to 
be preferred, as Leverrier maintained in 1870. It is hardly likely that 
transits will ever again be observed so elaborately and expensively. 


473. Gravitational Methods. — The scope of our work makes Gravita- 


it impossible to give any more than a very elementary explana- ^^^"^ 

tion of the principles involved, since the details of investiga- 
tion belong to a higher range. Of the different methods of this 
class we mention two only: 

(1) By the moon's parallactic inequality^ so called because by By parai- 
it the sun's parallax can be determined. lactic iu- 

It depends upon the fact that the sun's disturbing effect upon ^^^^ moou. 
the moon is sensibly greater in the half of the moon's orbit 
nearest the sun (^.e., the quarter on each side of new moon) than 
it is in the remoter half; and the difference depends upon the 
ratio between the radius of the moon's orbit around the earth to that 
of the eariKs orbit around the sun. If that ratio can be deter- 
mined, the radius of the earth's orbit comes out in terms of the 
distance of the moon from the earth, which is accurately known. 

As a consequence of this difference of the sun's disturbing 
force on the two halves of the orbit, the moon at the end of the 
first quarter is about two minutes of arc (120'') behind the 
place she would occupy if there were no such inequality in 
the disturbing force. At the third quarter (a week after full 
moon) she is as much ahead. 

If the moon's place could be observed as accurately as that of a star, 
this method would stand extremely high for precision ; but the observa- 
tional difficulties are serious, and the difficulty is much increased by the 
fact that at the first quarter we are obliged to observe the ivestern limb of 
the moon, and at the third the eastern. Still, the result obtained from it 
agrees very well with that from the other methods. 

474. (2) The second method is by the perturbations produced By pertur- 
by the earth on the orbits of Venus and Mars (and we may now ^^tionspro- 

*; . . duced by 

add Eros). The method depends upon the principle that the the earth. 
amount of these perturbations depends upon the ratio of the 
mass of the earth [including the moon) to that of the sun; and, 
further, that when this ratio of masses is known the distance 
of the sun follows at once from a simple equation, easily deduced. 



From Sec. 381, equation (1), we have, clianging a few letters 
(putting (;S'+ ^^) for (M+ m) and D for r), 

{S + E) = 





of parallax 
from the 
the masses 
of earth 
and snn. 

in which aS^ and E are the masses of the sun and earth, G is the 
constant of gravitation, D is the mean distance of the earth 
from the sun, and T the number of seconds in a year. 

Also, for the force of gravity at the earth's surface, we have 

E a 

g = G ^ —^ whence E = — x ?'^ r being the radius of the earth. 

Dividing the preceding equation by this, we get 

S + E _ 


L^ = 

If we put — = ili, this becomes D^ = ( — — ~ \gTh^% in which 
E \^ 4 TT" y ' 

everything is known if 31 is determined, g being given by pen- 
dulum observations and r by measurements of the earth's 
dimensions, while T is the length of the sidereal year in seconds. 
The matter can also be treated differently, bringing out tlie 
sun's distance in terms of the distance and pe?'iod of the moon, 
instead of g and r. 

The great beauty of the gravitational method lies in this, — 
that as time goes on and the effects of the earth upon the nodes 
and apsides of the neighboring orbits accumulate, the determina- 
its precision ^j^^^-^ q£ ^]^g earth's mass in terms of the sun's becomes continually 

cumulative. . . 

and cumulatively more precise. Even at present the method 
ranks high for accuracy, — so high that Leverrier, who first 
developed it, would, as already mentioned, have nothing to do 
with the transit-of- Venus observations in 1874, declaring that 
all such old-fashioned ways are absolutely valueless. By this 
method Newcomb deduces a parallax of 8". 768 ± 0'',010. 

of the gravi- 


475. It is to be noticed that the geometrical methods give Accordance 
the parallax of the sun directly^ apart from all hypothesis or of results 
assumption, except as to the accuracy of the observations them- the different 
selves, and of their necessary corrections for refraction, etc. The "lethods. 
gravitational methods, on the other hand, assume the exactness 
of the law of gravitation ; and the physical method (by the 
velocity of light) assumes that light travels in space with the 
same velocity as in our terrestrial experiments, after allowing 
for the retardation due to the refracting power of the air. The 
near accordance of the results obtained by the different methods 
shows that these assumptions must be very nearly correct, though 
perhaps not absolutely so. 

We add a little table giving the distance of the sun corre- Distance of 
spondino^ to different values of the solar parallax, assuming the ^^"^ corre- 

. . ^ . sponding to 

equatorial radius of the earth to be 3963.3 miles. different 

8''.75 corresponds to 23573 equatorial radii of the earth = 93 428000 miles, ^^l^^^f of the 

^ parallax. 

8^80 " " 23439 " " " " '^ = 92 897000 " 

8''.85 " " 23307 . " " " " " = 92 372000 " 

Newcomb, in his Astronomical Constants (1896) adopts 8". 797 
±0''.007 as the value of the solar parallax to be used in the 
planetary tables of the American ephemeris. 

He also gives the following as the results derived by the various methods Newcomb's 

after maJcing alloicance for probable systematic errors,' and assigns to each summary 

result the weight indicated by the number that follows it. °^ parallax 

Motion of the Node of Venus 8".768, 10 

GilVs Observations of Mars (1877) 8 .780, 1 

Pulkowa Constant of Aberration (20".-492) ... 8 .793, 40 

Contact Observations of Transit of Venus .... 8 .794, 3 

Heliometer Observations of Victoria and Sappho . . 8 .799, .5 

Parallactic Inequality of the Moon 8 .794, 10 

Miscellaneous Determinations of Aberration (20" AQ^^ 8 .806, 10 

Lunar Inequality of the Earth 8 .818, 1 

Measures of Venus in Transit 8 .857, 1 

Harkness, in his Solar Parallax and its Related Constants 
(1891), obtained as his final value 8".809 ± 0'^006. 



of comets. 


Their Number, Designation, and Orbits — Their Constituent Parts and Appearance — 
Their Spectra — Physical Constitution and Behavior — Danger from Comets 

476. The comets are bodies very different from the stars and 
planets. They appear from time to time in the heavens, remain 
visible for some weeks or months, pursue a longer or shorter 
path, and then fade away in the distance. They are called 
comets (from coma, i.e., "hair"), because when one of them is 
bright enough to be seen by the naked eye it looks like a star 
surrounded by a luminous fog and usually carries with it a 
long stream of hazy light. 

Large comets are magnificent objects, sometimes as bright 
as Venus and visible by day, with a dazzling nucleus and a 
nebulous head as large as the moon, accompanied by a train 
which extends half-way from the horizon to the zenith, and 
sometimes is really long enough to reach from the earth to the 
sun. Such are rare, however ; the majority are faint wisps of 
light, visible only with the telescope. 

Fig. 158 is a representation of Donati's comet of 1858, one 
of the finest ever seen. 

In ancient times comets were always regarded with terror, either as 
actually exerting malignant influences, or at least ominous of evil, and the 
notion still survives in certain quarters, although the most careful research 
fails to show, or even suggest, the slightest reason for it. 

477. Number of Comets. — Thus far, up to the beginning of 
the new century, our lists contain nearly eight hundred, about 
four hundred of Avliich were observed before the invention of 




the telescope in 1609, and therefore must have been bright. Of 
those observed since then, only a small proportion have been 

Fig. 158. — Naked-Eye View of Donati's Comet, Oct. 4, 1858 


conspicuous to the naked eye, — perhaps one in five. During 
the first half of the present century there were nine of this 



rank, and in the last half four, five of which were notable. 
The last really brilliant comet appeared in 1882. 

Since then there have been four or five that could be seen 
without a telescope, but only one bright enough to attract the 
notice of the casual observer, — Rordame's comet of 1893. In 
August, 1881, for a few days two conspicuous comets — one 
magnificent one, and the other more than respectable — shone 
together in the northern heavens not very far apart, a thing 
almost, if not quite, unprecedented. 

The total number that visit the solar system must be enor- 
mous, since, although even with the telescope we can see only 
the comparatively few which come near the earth and are favor- 
ably situated for observation, yet not infrequently from five to 
eight are discovered in a single year (ten in 1898); and there 
is seldom a day when one is not present somewhere in the sky : 
often there are several. 
Methods of 478. Designation of Comets. — A remarkable comet gener- 
designating ^lly bears the name of its discoverer or of some one who has 
acquired its " ownership," so to speak, by some important 
research respecting it. Thus, we have Halley's, Encke's, and 
Donati's comets. The common herd are distinguished only 
by the year of discovery, with a letter indicating the order of 
discovery in that year, as comet a, 5, (?, 1895 ; or, still again, by 
the year, with a Roman numeral denoting the order of perihelion 
passage. Thus, Donati's comet, which is " comet /, 1858," is 
also " comet 1858-VI," and this is the more scientific designa- 
tion, and is generally used in catalogues of comets. 

Comet a is not, however, always comet I, for comet h may 
outrun it in reaching the perihelion, and it often happens that 
a comet's perihelion passage does not occur in the same year 
as its discovery. 

In some cases a comet bears a double name, as, for instance, 
the Pons-Brooks comet, which was first discovered by Pons in 
1812, and on its return in 1883, by Brooks. 



479. Discovery of Comets. — As a rule, these bodies are first Their 
found by comet hunters, who make a business of searching for ^^^'^^ery. 
them. For this purpose they usually employ a telescope known 

as a " comet-seeker," carrying an eyepiece of low power, with a 
large field of view. When first seen a comet is usually a mere 
roundish patch of faintly luminous cloud, which, if really a 
comet, will reveal its true character within an hour or two by 
its motion. 

Some observers have found a great number of these bodies. Messier 
discovered thirteen between 1760 and 1798, and Pons twenty-seven between 
1800 and 1827. In this country Brooks, Barnard, and Swift have been 
especially successful. It occasionally happens, however, as with Holmes' 
comet of 1892, and Rordame's comet of 1893, that a comet is picked up 
with the naked eye by some one not an astronomer at all. 

Recently three have been discovered by photography, the first by Bar- 
nard at the Lick Observatory in 1892, the second by Chase in 1898 while 
trying to photograph November meteors, and the third by Coddington in 
1899, also at the Lick Observatory. 

480. Duration of Visibility, and Brightness. — The comet of Duration of 
1811 was observed for seventeen months ; the great comet of "^'^^^^^^^^5'- 
1861 for a year; and comet 1889-1 was followed at the Lick 
Observatory for nearly two years, — the longest period of visi- 
bility yet recorded. On the other hand, the comet is some- 
times visible only a week or two, and twice a comet near the 

sun has been photographed during a total eclipse, — never seen 
before or after the two minutes of totality. The average dura- 
tion of visibility is probably not far from three months. 

As to brightness, comets differ widely. About one in four or Their 
five reaches the naked-eye limit at some point in its orbit, and a 
very few, say two or three in a century, are bright enough to be 
seen in the daytime. The comets of 1843 and 1882 were the 
last so observed. 





ideas as 
to their 

Tycho estab- 
lishes their 

of parabolic 
orbits by 

numbers of 
and hyper- 
bolic orbits. 

481. The ideas of the ancients as to the motions of these 
bodies were very vague. Aristotle and his school considered 
them to be merely exhalations from the earth, inflamed in the 
upper air, and therefore meteorological bodies, and not astro- 
nomical at all. Seneca, indeed, held a more correct opinion, 
but it was shared by few ; and Ptolemy fails to recognize them 
as heavenly bodies in his Almagest. 

Tycho was the first to establish their rank as truly " celes- 
tial," by comparing the observations of the comet of 1577, made 
in different parts of Europe, and showing that its parallax was 
less, and its distance greater, than that of the moon. 

Kepler supposed that they moved in straight lines and seems 
to have been more than half disposed to consider them as living 
beings, traveling through space with will and purpose, "like 
fishes in the sea." 

Hevelius in 1675 was the first to suggest that their orbits 
might be parabolas, and his pupil Doerfel proved this to be the 
case in 1681 for the comet of that year. The theory of gravi- 
tation had now appeared, and Newton soon worked out and 
published a method by which the elements of a comet's orbit 
can be determined from the observations. 

482. Relative Numbers of Parabolic, Elliptical, and Hyperbolic 
Orbits. — A large majority move in orbits that are sensibly parab- 
olas. Out of nearly four hundred orbits computed up to 1901, 
more than three hundred are of this kind. About eighty-five 
are more or less distinctly elliptical^ and about half a dozen 
seem to be hyperbolas., but hyperbolas differing so slightly from 
the parabola that the hyperbolic character is not certain in a 
single one of the cases. 

Comets which have elliptical orbits of course return, if 
undisturbed, at regular intervals ; the others visit the sun 
only once, and never come back. 



The difficulty of determining whether a particular comet is Difficulty of 

or is not periodic is much increased by the fact that comets ^ecogmzmg 

^ ^ ^ -^ a comet 

have no characteristic " personal appearance," so to speak, by when it 

which a given individual can be recognized whenever seen, — returns. 
as Jupiter or Saturn could be, for instance. It is necessary to 
depend almost entirely upon the elements of its orbit for the 

Fig. 159. — The Close Coincidence of Different Species of Cometary Orbits 

within the Earth's Orbit 

recognition of a returning comet, and this is not always satis- 
factory, since there are a number of cases in which several 
distinct comets move in orbits almost identical. (See Sec. 487.) 
483. Elements of a Comet's Orbit. — As in the case of a 
planet, three perfect observations of a comet's place are theo- 
retically sufficient to determine its entire orbit. Practically, mined by 
however, it is not possible to observe a comet with anything ^^^^^^ ^°^" 

,., , /> 1 f ' • -\ r- ' plete obser- 

like the accuracy oi a planet (on account oi its mdennite out- vations. 
line), nor usually v/ith sufficient exactness to determine positively 

Elements of 
a comet's 
orbit deter- 



orbit has 
but five 

from a small number of observations whether the orbit is or 

is not parabolic. 

The plane of the orbit and its perihelion distance can, in most 
Uncertainty cascs, be fairly settled without any difficulty; but the eccen- 
as to major f^{(.{iy ^nd the major axis, with its corresponding period, require 

axis and . . n i • i ■ i 

period. a long Series of observations for their determination and are 

seldom ascertained with much precision from a single appear- 
ance of the comet. In that part of the comet's path which can 
be observed from the earth the three kinds of orbits usually 
diverge but little ; indeed, they may almost coincide (as shown 
in Fig. 159). 

For a parabolic orbit the elements to be computed are only 
five in number, instead of seven, as in the case of an ellipse. 
The semi-major axis and period (which are infinite) drop out, 
as does the eccentricity/, which is necessarily unity. To define 
the size of the orbit the perihelion distance, p, takes the place 
of the semi-major axis. 

For the parabolic elements we have, therefore, (1) p, peri- 
helion distance, (2) i, inclination of the orbit to the ecliptic, 
(3) Q, , the longitude of the ascending node, (4) co, angle 
between line of nodes and perihelion, (5) T, date of perihelion 

It must be distinctly understood, moreover, that orbits which are 
" sensibly " parabolic are seldom, if ever, strictly so, — the chances are 
infinity to one against an exact parabola. If a comet were moving at any 
time in such a curve, the slightest retardation due to the disturbing force 
of any planet would change this parabola into an ellipse, and the slightest 
acceleration would make an hyperbola of it. 

Effect of It should be noted also that if a comet's orbit is nearly para- 

siight change ^qI[q ^ very slis^ht chansce in the velocity of the comet's motion 

of comet s ^ .y o o ^ ^ 

velocity upon will causc ail eiiomious change in the computed major axis and 
major axis period. This is obvious from the equation (Sec. 320) 

and period in ^ 

cases of _ ^ /^ ^^ A > 

orbits nearly ^ 9 l /T2 7/2 ) * 

parabolic, ^ ^ 



When V nearly equals U (as must be the case if the orbit is 
nearly a parabola) the denominator will be extremely small, 
and a very trilling change in V will make a great percentage of 
change in the difference between V^ and F^, and will affect a, 
the semi-major axis, accordingly. 

484. The Elliptic Comets. — The first comet ascertained to 
move in an elliptical orbit was that known as Halley's, which 
has a period of about seventy-six years, its periodicity having 
been discovered by Hal- 
ley in 1681. It has since 
been observed in 1759 
and 1835 and is due 
again about 1911. 

The second of the 
periodic comets in order 
of discovery is Encke's, 
with the shortest period 
known, less than three 
and one-half years. Its 
periodicity was discov- 
ered in 1819. 

About a dozen of the 
comets to which compu- 
tation assigns elliptic 
orbits have periods so 

long — near or exceeding one thousand years — that their real 
character is still rather doubtful. About seventy-five, however, 
have orbits which are distinctly and certainly elliptical, and 
about sixty of them have periods of less than one hundred 
years. About twenty have been actually observed at two or 
more returns to perihelion; as to the rest of the sixty, several 
are now expected within a few years, and many have prob- 
ably been lost to observation, either from disintegration, like 
Biela's comet (soon to be discussed), or by having their orbits 

Fig. 160. — Orbits of Short-Period Comets 




comets of 
return has 



Relation of 
comets to 

families ; 
origin of 

The comet- 
families of 
Uranus, and 

The capture 

transformed by perturbations, so that they no longer come 
within the range of observation. 

Fig. 160 shows the orbits of five of the short-period comets 
(as many as can be shown without confusion) and also a part 
of the orbit of Halley's comet. These five particular comets, 
and about twenty-five more, all have periods ranging from three 
and one-half to eight years, and they all pass very near the 
orbit of Jupiter. Moreover, each comet's orbit crosses that of 
Jupiter near one of its nodes, marked by a short cross line on 
the comet's orbit. The fact is extremely significant, showing 
that these comets at times come very near to Jupiter, and it 
points to an almost certain connection between that planet and 
these bodies. 

485. Comet-Families ; Origin of Periodic Comets. — It is clear, 
as has been said, that the comets which move in parabolic orbits 
cannot well have originated within the limits of the solar system, 
but must have come from a great distance. As to those which 
move in elliptical orbits, it is a question whether we are to 
regard them as native to the system or only as "naturalized," 
or perhaps mere sojourners for a time ; but it is evident that in 
some way many of them stand in peculiar relations to Jupiter 
and to other planets. 

The short-period comets, those which have periods ranging 
from three to eight years, are now recognized and spoken of as 
Jupiter's /am^7«/ of comets. About thirty are known already, of 
which fifteen have been observed twice or oftener, — some of 
them a dozen times. Similarly, Saturn is credited with two 
comets; Uranus with two, one of which is Tempel's comet, 
closely connected with the November meteors and due to appear 
in 1900, but not seen. Finally, Neptune has a family of six; 
among them Halley's comet, and two others which have returned 
a second time to perihelion since 1880. 

486. The Capture Theory. — The now generally accepted expla- 
nation as to the origin of these cometfamilies was first suggested 


by Laplace; viz., that the comets which compose them have been 
"captured" by the planet to which they stand related. 

A comet entering the system in a parabolic orbit and pass- 
ing near the planet will be disturbed and either accelerated or 
retarded. If it is accelerated, then according to equation (4) 
(Sec. 320), the major axis will become negative, the orbit will Effect of 
be chang^ed to an hyperbola, and the comet will never be seen I'^^^^^^^^^'i 

^ . . . , . ^^^' accelera- 

again. But if the comet is retarded, the semi-major axis will tiou to 
become finite and the orbit will be made elliptical, so that the trausform a 


comet will return at each revolution to the place where it was orbit. 
first disturbed ; it will become a j^^'^^odic comet, with its orbit 
passing near to the orbit of the disturbing planet. 

It will not, however, as students sometimes imagine, revolve 
around its capturer like a satellite. The focus of its new and 
diminished orbit still remains at the sun. 

But this is not all. After a certain time the planet and the Subsequent 
comet will be sure to come together ag-ain at or near this place, ^u^ounters 

*=" ... of comet 

The result then 7na^ be an acceleration which will enlarge the and planet, 
comet's orbit, or even transform it to a parabola or hyperbola ; 
but it is an even chance at least that the result may be a retarda- 
tion and that the orbit and period may thus be further dimin- 
ished. This may happen over and over again, until the planet's 
orbit falls so far inside that of the planet that it suffers no 
further disturbance to speak of. 

Given time enough and comets enough, the ultimate result 
would necessarily be such a comet family as really exists. It 
is not permanent, however ; sooner or later, if a captured comet 
is not first disintegrated, it will almost certainly encounter its 
planet under such conditions as to be thrown out of the system. 

A recent investigation, however, by Callandreau, upon the Caiian- 
disintcQTation of comets by the action of the sun and the ^^"^'^^^.^ 

^ -^ luvestiga- 

planet Jupiter, shows that the limit of distance at which such tion on the 
an effect is possible is quite considerable, and that the breaking^ ^I'eakmgup 

^ of comets. 

up of a comet ought not to be very unusual. He suggests that 




Members of 
a comet- 
group have 
a common 
origin; to 
be care- 
fully dis- 
from comet- 

distance of 

the number of the comets of Jupiter's family has probably thus 
been largely increased by the division of single comets into 
several, — a suggestion which greatly relieves very serious 
objections that have been urged against the capture theory. 

487. Comet-Groups. — There are several instances in which a 
number of comets, certainly distinct, chase each other along 
almost exactly the same path, at an interval usually of a few 
months or years, though they sometimes appear simultaneously. 
The most remarkable of these comet-groups is that composed 
of the great comets of 1668, 1843, 1882, and 1887. These 
have all come in from the direction (nearly) of Sirius and have 
receded nearly on that line, passing close around the sun and 
actually through the corona. As Professor Comstock has pointed 
out, they are all of them, if their computed orbits can be trusted, 
now (1902) bunched together in a space hardly bigger than the 
sun, at a distance of about 150 radii of our orbit, and are moving 
away together very slowly. 

It is, of course, nearly certain that the comets of such a group 
have a common origin, perhaps from the disruption of a single 
comet by the attraction of the sun or a planet, in accordance 
with the suggestion of Callandreau just mentioned. 

The distinction between covaQt-families and GOVnQt-groups 
must be carefully noted : in the former the orbits agree only in 
passing close to that of the capturing planet; in the latter the 
orbits are nearly identical, at least in the part near the sun. 

488. Perihelion Distance, etc. — The perihelion distmices of 
comets differ greatly. Eight of the 300 computed orbits 
approach the sun within less than 6 000000 miles, and four 
have a perihelion distance exceeding 200 000000. A single 
comet, that of 1729, had a perihelion distance of more than 
four astronomical units, or 375 000000 miles; this is one of the 
half dozen possibly hyperbolic comets, and must have been an 
enormous one to be visible at such a distance. There may, 
of course, be any number of comets with still greater perihelion 


distances, because, as a rule, we are able to see only such as 
come reasonably near to the earth's orbit, — probably but a 
small percentage of the total number that visit the sun. It 
has been computed that something like six thousand come 
within the orbit of Jupiter every year. 

The inclinations of cometary orbits range all the way from inclination 
zero to 90°, but most of the short-period comets have orbits ^ ^^^^ ^^^ 
of small inclination, as might be expected, since such comets 
would be much more likely to suffer capture than those that 
cross the planes of the planetary orbits at a high angle. 

As regards the direction of motion^ the six hyperbolic comets Direction of 
and all the elliptical comets having periods of less than one "^*^*^^^- 
hundred years move direct^ excepting only Halley's comet and 
Tempel's comet of 1866. The rest show no decided preponder- 
ance either way. 

489. Comets are Visitors. — The fact that the orbits of most Comets are 
comets are sensibly parabolic, and that their planes have no evi- ^^^^^^^'^ ^^ 

^ . the solar 

dent relation to the ecliptic, apparently indicates (though it does system, 
not absolutely demonstrate) that these bodies do not in any 
proper sense belong to the solar system. They come to us with 
just the velocity they would have if falling towards the sun 
from an enormous distance, and they leave the system with a 
velocity which, if no force but the sun's attraction acts upon 
them, will carry them away to an infinite distance, or until they 
encounter the attraction of some other sun. 

Their motions are just what might be expected of ponderable 
masses moving in empty space between the stars under the law 
of gravitation. 

There are difficulties with the theory that the comets come to Difficulty 
us from space among the stars, chiefly depending: upon the now ^^^^^ ^^^' 

^ ^ ' J r & r pothesisthat 

certain fact that the solar system is traveling at the rate of they come 
several miles a second (Sec. 543) and that, therefore, comets f^'o^n stellar 


composed of matter met by us ought to have a relative velocity, 
with respect to the sun, so great as to produce numerous 



The home 
of the 

behavior of 

increase of 
speed as 
result of 

hyperbolic orbits, whereas we find few such, if any. Then, too, 
there ought to be a marked concentration of the axes of cometary 
orbits near the direction towards which the sun is moving. 

While the investigations of the late Professor Newton of 
New Haven partially relieve the difficulty, astronomers still 
feel it; and many are disposed to think that our solar system, 
in its journey through space, is accompanied by far-distant, 
outlying clouds of nebulous matter, which are the source and 
original "home of the comets," to borrow Professor Peirce's 

490. Acceleration of Encke's Comet. — With one remarkable 
exception, the motions of comets appear to be just what would 
be expected of masses moving in free space under the law of 
gravitation. The single exception is in the case of Encke's 
comet, which, since its first discovery in the last century (its 
periodicity was not discovered until 1819), has been continu- 
ally quickening its speed and shortening its period. In 1819 
its period w^as 1205 days. Between 1820 and 1860 each suc- 
cessive period shortened about two and one-half hours ; from 
1860 to 1870 the shortening was only one and three-fourths 
hours to each revolution, and since then it has increased to 
about two hours. The period at present is about fifty-four 
hours shorter than in 1819, and the mean distance from the 
sun is nearly a quarter of a million of miles less than then. 

No perturbations by any known body will account for such 
an acceleration, and thus far no reasonable explanation has been 
suggested as even possible, except that something encountered 
in its motion through interstellar space retards the comet, just 
as air retards a rifle bullet. 

At first sight it seems almost paradoxical that a resistance 
should accelerate a comet's speed, but referring to Sec. 320 we 
see that any diminution of the velocity will also diminish the 
semi-major axis. This will reduce the period, which is propor- 

tional to V a^ by a greater percentage than it will reduce the 


circumference of the orbit, which is simply proportional to a; as 
a consequence there will be an increase of velocity above what 
the comet had in the larger orbit. A comet gains more speed 
hi/ falling nearer to the sun than it loses by the direct effect of 
the resistance. If this action continues without cessation, the 
ultimate result must be a spiral winding inward until the comet 
strikes the surface of the sun. 

When this peculiar behavior was first discovered by Encke it was 
ascribed to the action of a resisting medium and adduced as proof of 
the existence of the " luminif erous ether." But since no other comets 
exhibit the same effect, and the effect upon Encke's comet itself varies in 
amount from time to time, it is now generally attributed to something 
encountered along the orbit of this particular body ; possibly the passage 
through some cloud of meteors, or disturbances by some unknown body in 
the asteroidal regions. 


491. Physical Characteristics of Comets. — The orbits of these Physical 
bodies are now thoroughly understood, and their motions are 
calculable with as much accuracy as the nature of the observa- 
tions permit; but we find in their physical constitution and 
behavior some of the most perplexing and baffling problems in 
the whole range of astronomy, — apparent . paradoxes which 
have not yet received a satisfactory explanation. 

While comets are evidently subject to the attraction of gravi- 
tation, as shown by their orbits, they also exhibit evidence of 
being acted upon by powerful repulsive forces emanating from 
the sun. While they shine partly by reflected light, they are 
also certainly self-luminous, their light being generated in a way Non- 
not yet thoroughly explained. They are the bulkiest bodies P'^^^tary 
known, except the nebulse, in some cases thousands of times ities. 
larger than the sun or stars; but in mass they are "airy noth- 
ings," and one of the smaller asteroids probably livals the 
largest of them in weight. 




The coma. 

The nucleus. 

The train, 
tail, or 

Jets and 

of heads of 

492. The Constituent Parts of a Comet. — (a) The essential 
part of a comet — that which is always present and gives the 
comet its name — is the coma^ or nebulosity, a hazy cloud of 
faintly luminous transparent matter. 

{h) Next, we have the nucleus^ which, however, makes its 
appearance only when the comet is near the sun, and is wanting 
in many comets. It is a bright, more or less starlike point 
near the center of the coma, and is usually the object " observed 
on " in noting a comet's place. In some cases the nucleus is 
double, or even multiple. 

[c) The tail^ or train^ is a stream of light which commonly 
accompanies a bright comet and is sometimes present even with 
a telescopic one. As the comet approaches the sun the tail 
follows it, but as the comet moves away from the sun it pre- 
cedes, and by the ancients was then called the heard. Speaking 
broadly, the train is always directed away from the sun, though 
its precise form and position are determined partly by the comet's 
motion. It is practically certain that it consists of extremely 
rarefied matter, which is thrown off by the comet and powerfully 
repelled by the sun. It certainly is not — like the smoke of a 
locomotive or the train of a meteor — matter simply left behind. 

{d) Jets and Envelopes. The head of a brilliant comet is 
often veined by jets of light, which appear to be spirted out 
from the nucleus ; and sometimes it throws off a series of concen- 
tric envelopes like hollow shells, one within the other. These 
phenomena, however, are seldom observed in any but brilliant 

493. Dimensions of Comets. — The volume, or bulk, of a 
comet is often enormous, — almost beyond conception if the 
tail is included in the estimate. The head, or coma, is usually 
from 40000 to 150000 miles in diameter; a comet less than 
10000 miles in diameter would stand little chance of discovery, 
and comets exceeding 150000 miles are rather unusual, though 
there are a considerable number on record. 


The head of the comet of 1811 at one time measured nearly 1 200000 
miles, — more than forty per cent larger than the diameter of the sun itself. 
Holmes' comet of 1892 had at one time a diameter exceeding 700000 miles, 
but no visible nucleus at that time. A few weeks later it looked like a 
mere hazy star. The comet of 1680 had a head 600000 miles across, and 
that of Donati's comet of 1858 was 250000 miles in diameter. 

The diameter of the head changes all the time, and what is 
singular is, that while the comet is approacliing the sun, the 
head ordinarily contracts^ expanding again as it recedes. The Contraction 
diameter of Encke's comet shrinks from about 300000 miles ^^ ^^^^^^ 

when near 

when it is 130 000000 miles from the sun to a diameter not the sun. 
exceeding 12000 or 14000 miles when at perihelion, a distance 
of 33 000000 miles, the variation in bulk being more than 
10000 to 1. No satisfactory explanation is known, but Sir 
John Herschel has suggested that the change may be merely 
optical, — that near the sun a part of the nebulous matter is 
evaporated by the solar heat and so becomes invisible, con- 
densing and reappearing again when the comet reaches cooler 

The nucleus usually has a diameter ranging from a mere point Diameter of 
less than 100 miles in diameter up to 5000 or 6000, or even ^^^^ nucleus. 
more. Like the comet's head, it also changes in diameter, even 
from day to day. The variations, however, do not seem to 
depend in any regular way upon the comet's distance from 
the sun, but rather upon its activity in throwing off jets and 

The tail of a comet, as regards simple magnitude, is by far its Dimensions 
most imposing feature. Its length is seldom less than 5 000000 °^ ^^^® tr&m. 
or 10 000000 miles ; it frequently attains 50 000000, and there 
are several cases in which it has exceeded 100 000000. It is 
usually more or less fan-shaped, so that at the outer extremity 
it is millions of miles across, being shaped roughly like a cone 
projecting behind the comet from the sun, and more or less bent its usual 
lik© a horn, as shown in Fig. 158. The volume of the train of ^°^""^" 



Mass of 

Nature of 
the evi- 

May equal 
mass O'f an 
iron ball 
150 miles in 

the comet of 1882, 110 000000 miles in length, some 200000 
miles in diameter at the comet's head, and with a diameter 
of 10 000000 or 12 000000 at its extremity, exceeded the bulk 
of the sun itself more than eight thousand times. 

494. Mass of Comets. — While the volume of comets is thus 
enormous, their mass is apparently insignificant, — in no case 
at all comparable even with that of our little earth. 

The evidence on this point, however, is purely negative ; it 
does not enable us in any case to determine how great the mass 
really is, but only how great it is not; ^.g., it only proves that 
the comet's mass is less than a certain very small fraction of the 
earth's, but does not warrant us in setting any lower limit. 

The evidence is derived from the fact that no sensible pertur- 
bations have ever been produced in the motions of the planets 
or their satellites even when comets have come very near them ; 
and yet in such a case the comet itself is "sent kiting" in a 
new orbit, showing that gravitation is fully operative between 
the comet and the planet. 

Lexell's comet in 1770, and Biela's comet on several occa- 
sions, came so near the earth that the length of the comet's 
period was greatly changed, while the year was not altered by 
so much as a single second ; and it would have been changed 
by many seconds if the comet's mass were as much as ^ o^-^o"o o^ ^^ 
that of the earth. 

Brooks' comet of 1886 actually passed between Jupiter and 
the orbit of its first satellite. None of the satellites were 
sensibly disturbed, but the comet's orbit was changed from an 
ellipse with a period of over thirty years to one of a period 
with less than seven. 

At present this mass (^ o qVo o^ ^^ ^^^ earth's mass) is very gen- 
erally assumed as a probable upper limit for even a large comet. 
It is about ten times the mass of the earth's atmosphere and is 
about equal to the mass of a ball of iron 150 miles in diameter, 
but how much smaller the limit may really be no one can say. 


495. Density of Comets. — The mean density is necessarily Mean 
extremely low, the mass of the comet being so small and the ^®^^^^y ^^ 

•^ ^ _ comets very- 

volume SO great. If the head of a comet 50000 miles in low. Com- 

diameter has the very improbable mass of tttttVtttt of that of the payable with 

. .., ,-^Vci ri^^ air-pump 

earth, its mean density is only about g qVo P^^^ ^^ ^"-'^^ ^^ ^'^^ vacuum. 
air at the earth's surface, — a degree of rarefaction reached by 
only the very best air-pumps. 

The extremely low density of comets is shown also by their 
transparency. Small stars are often seen directly through the Transpar- 
head of a comet 100000 miles in diameter, even very near its ^^^^^^ 

^ comets. 

nucleus, and with hardly a perceptible diminution of luster. 
There are, however, in such cases indications of a very slight 
refraction of the light passing through the comet, causing a 
barely sensible displacement of the star. 

As for the tail, the density of this must be almost infinitely stm lower 
lower than that of the head, — far below the best vacuum we can ^^^^^^y ^^ 

the tram. 

make by any means of science. It is nearer to an airy nothing 
than anything else we know of. 

Another point should be referred to. Students often find it 
hard to conceive how such impalpable " dust clouds " can move Dust clouds 
in orbits like solid masses and with such enormous velocities ; ^^'^^^^i'^® 


they forget that in a vacuum a feather falls as swiftly as a stone, etary space 
Interplanetary space is a vacuum, far more perfect than any- as swiftly 
thing we can produce by artificial means, and in it the lightest ][)odies. 
bodies move as freely and swiftly as the densest, since there is 
nothing to resist their motion. If all the earth were suddenly 
annihilated, except a single feather, the feather would keep 
on and pursue the same orbit, with the unchanged speed of 
18^ miles a second. 

496. Nature of Comets. — We must bear in mind, however. Low mean 
that the low mean density of the comet does not necessarily jnco^^ati^ 
imply that the density of its constituent parts is small. A biewith 
comet may be in the main composed of small heavy bodies and l^ ensity 

■^ ^ «^ of constitu- 

still have a very low mean density, provided they are widely ent parts. 



swarms of 
small solid 

The light of 
comets not 
largely due 
to action of 
solar rays. 


The ordi- 
nary comet 

enougli separated. There is much reason, as we shall see, foi 
supposing that such is really the case, — that the comet is 
largely composed of small meteoric sand grains (say pinheads, 
many feet apart), each carrying with it a certain quantity of 
enveloping gas, in which light is produced either by electric 
discharges or by some different action due to the rays of 
the sun. 

As to the size of the particles opinions vary widely: some 
maintain that they are large rocks ; Professor Newton calls a 
comet a " gravel bank " ; others think it a mere '' dust cloud " 
or "smoke wreath." 

The unquestionable and close connection between comets and 
meteors, which we shall soon discuss, almost compels some 
"meteoric hypothesis," and, at present at least, no other theory 
is maintained by any high authorities. 

497. The Light of Comets. — To some extent this is reflected 
sunshine, but in the main it is light emitted by the comet itself 
under the stimulus of solar action. That the light depends in 
some way upon the sun is shown by the fact that its intensity 
follows approximately the same law as the brightness of a 

planet, and is usually proportional to , in which R is the 

comet's distance from the sun and A its distance from the earth. 

A comet as it recedes from the earth does not simply grow 
smaller, retaining the same apparent intrinsic brightness, as 
would be the case with an independently self-luminous body, 
but grows fainter and disappears on account of faintness. 

Not infrequently, however, the light of a comet varies capri- 
ciously, brightening and fading without apparent cause, within 
a few days or even a few hours. 

498. Spectra of Comets. — The spectrum is usually a faint 
continuous spectrum, on which are superposed certain bright 
bands, five of them in the visible spectrum ; there are others 
in the ultra-violet, observable only by photography. Of the 



five visible bands, two are very faint, so ttiat ordinarily but 
three can be seen. Tlie spectrum is identical with that of the 
blue cone at the base of a Bunsen-burner flame, which is always 
found where hydrocarbon gases are in a state of combustion, 
and is generally ascribed to acetylene. (See Fig. 161, comet, 
1881-III.) The spectrum of a comet is not mainly, as some- 

FiG. 161. — Comet Spectra 

For convenience in engraving, the dark lines of the solar spectrum in the lowest 
strip of the figure are represented as bright 

times stated in text-books, the spectrum of carhon monoxide, 
though some slight peculiarities in the comet's spectrum sug- 
gest the presence, at times, of this gas also. 

The faint continuous spectrum is due, in part at least, to 
reflected sunlight, as shown by the fact that some of the prin- 
cipal Fraunhofer lines have been pliotographed in it, though 
they cannot be seen. 

If the nucleus is bright, its spectrum also appears like a 
narrow streak, nearly continuous, running through the spec- 
trum of the head, as shown in the figure. At least ninety per 




Bright lines 
in spectrum 
of 1882 -11. 

Question as 
to cause of 

cent of all the comets thus far observed have given this hydro- 
carbon (acetylene?) spectrum. 

If the comet is one that does not approach the sun within the 
distance of 100 000000 miles or so (such comets are not numer- 
ous), the hydrocarbon bands are sometimes missing, replaced in 
some cases by unidentified bands of a different wave-length, as 
in the case of Brorsen's comet and Borrelly's comet of 1877 
(Fig. 161). 

The spectrum of Holmes' comet of 1892, which never came 
inside the earth's orbit, showed no bands or lines at all, either 
bright or dark, but was simply continuous. 

If, on the other hand, the comet approaches the sun within 
8 000000 or 10 000000 miles, the hydrocarbon bands grow 
relatively faint, and the yellow line of sodium becomes domi- 
nant, as in Wells' comet, 1882-1, and the great comet, 

The latter, indeed, which almost grazed the surface of the 
sun, showed numerous bright lines of other substances (prob- 
ably iron for one). 

It has been maintained by Sir Norman Lockyer that the 
comet's spectrum changes regularly and progressively with dis- 
tance from the sun, the bands not only altering their appear- 
ance, but slightly shifting their position ; but the evidence for 
this is not conclusive. 

As to the cause of luminosity, it is practically agreed that it 
cannot be due to any general heating of the mass of the comet, 
of which the mean temperature, on the contrary, is probably 
extremely low. The explanation now most favored attributes 
the light to electric discharges between the solid (?) particles 
through the gases which envelop them, — discharges due to 
inductive action of the sun on the " cometic " cloud rushing 
towards it from regions of space, where the electric potential is 
presumably different from that of the sun itself. At present 
we can assign no certain reason for such difference, but, on the 




other hand, there is not any known reason for assuming 
uniform electric potential through all space. (See Sec. 502.) 

It is, perhaps, necessary to remark that while the hydrocarbon Comet not 
bands of the spectrum demonstrate the presence of hydrocarbons "^^i^ y^^om- 

■^ ^ "^ ^ ^ posed of 

in the comet, they do not at all prove that the comet is mainly hydro- 
composed of them, nor even that they constitute a considerable c^^'^^ns. 
portion of its mass. It is much more likely that the minute solid 
or liquid particles constitute ninety per cent of the whole. 

Fig. 162. — Head of Donati's Comet 

499. Phenomena that accompany the Comet's Approach to the Phenomena 
Sun. — When a comet is first discovered it is usually, as has ^'esuitmg 
been already said, a mere round nebulosity, a little brighter proachto 
near the middle. As it approaches the sun it brightens rapidly, *^e sun. 
and the nucleus appears. Then on the sunward side the 
nucleus appears to emit luminous jets, or to throw off more or 
less symmetrical envelopes, which follow each other at intervals 
of a few hours, expanding and growing fainter, until they are 
lost in the general nebulosity of the head. 



and jets. 

of tail by- 
and sun. 

A To Sun 

Fig. 162 shows the envelopes as they appear in the head 
of Donati's comet of 1858. At one time seven of them 
were visible at once ; very few comets, however, exhibit 
the phenomena with such symmetry. More frequently the 
emissions from the nucleus take the form of mere jets and 

500. Formation of the Tail. — The tail appears to be formed 
of material first projected from the nucleus towards the sun and 
afterwards repelled both by nucleus and sun, as illustrated by 

Fig. 163. At least, this theory 
has the great advantage over 
all others which have been pro- 
posed (there have been many of 
them) that it not only accounts 
for the phenomenon in a general 
way, but admits of being worked 
out in detail and verified mathe- 
matically, by comparing the 
actual size and form of the comet's 
tail at different points in the orbit 
with that indicated by theory ; 
Fig. 163. — Formation of a Comet's and the accordance is usually 

Tail by Matter expelled from the ^ . „ , 

Head satisfactory. 

According to this theory, the 
tail is simply an assemblage of repelled particles, each moving 
in its own hyperbolic orbit ^ around the sun, the separate 
particles having very little connection with, or effect upon, each 

1 Since the assumed repulsive force upon a particle virtually diminishes the 
sun's attraction upon it, it also virtually diminishes its parabolic velocity (i.e., 
if under this diminished attraction the particle had fallen from an infinite 
distance, its parabolic velocity would be less than if gravitation had acted 
unmodified). In the formula of Sec. 320, U^, if the comet is moving in a 
parabola, therefore becomes less than V^ for the particles that compose the 
tail ; and the semi-major axis, a, for the subsequent orbit of such particles, 
becomes negative^ converting their orbits into hyperbolas. 


other and being almost entirely emancipated from the control 
of the comet's head. 

Since the force of the projection from the comet is seldom 
very great, all these orbits lie nearly in the plane of the comet's 
orbit, and the result is that the tail is usually a sort of a flat, The tail 
hollow^ curved, horn-shaped cone, open at the large end. The ^suaiiy a 
edges of the tail, near the comet at least, therefore usually curved cone, 
appear much brighter than the central part. 

501. Curvature of the Tail, and Tails of Different Types. — 
The tail is curved, because the repelled particles, after leaving Explanation 

of curva- 

Fig. 164. — A Comet's Tail at Different Points in its Orbit 
near Perihelion 

the comet's head and receding from the sun, retain their origi- 
nal motion, and in consequence are arranged, not along a 
straight line drawn from the sun to the comet, but on a curve 
convex to the direction of the comet's motion, as shown in 
Fig. 164, — the stronger the repulsion, the less the curvature. 

Bredichin of Moscow has found that in this respect the trains The three 
of comets may be classified under three different types : ^^p^^ °^ 

T7' 1 7 '7 1 \ c comet's 

Firsts the long^ straight rays : they are composed of matter tails. 
upon which the solar repulsion is from twelve to fifteen times as 






The hydro- 
carbon tail. 

Tails due 
to metallic 

Nature of 
the repulsive 
force : the 

great as gravitational attraction, so that the particles leave 
the comet with a relative velocity of 4 or 5 miles a second, 
which, is afterwards continually increased until it becomes 
enormous. The nearly straight rays, shown in Fig. 158, tan- 
gent to the principal tail of Donati's comet, belong to this 
class. For plausible reasons, connected with the low density 
of hydrogen, Bredichin considers them to be composed of that 
substance, possibly set free by the decomposition of hydrocar- 
bons. They are rather uncommon, and in no case since the 
promulgation of the theory have been bright enough to allow a 
spectroscopic test of their nature. 

The second type is the curved, plumelike train, like the prin- 
cipal train of Donati's comet. In trains of this type, supposed 
to be due to hydrocarbon vapors, the repulsive force varies from 
2.2 times the gravitational attraction for particles on the convex 
edge of the train to half that amount for those on the inner 
edge. Trains of this class show the hydrocarbon spectrum 
through all their extent. 

Third. A few comets show tails of still a third type, — short, 
stubby brushes, violently curved, and due to matter upon which 
the repulsive force is feeble as compared with gravity. These 
are assigned to metallic vapors of considerable density, sodium 
perhaps, possibly sometimes iron. 

502. The Repulsive Force. — The nature of the force which 
repels the particles of a comet is, of course, only a matter of 
speculation. There is probably at present a decided preponder- 
ance of opinion in favor of the idea that it is electrical. In this 
case the repulsion upon small particles, being a surface action, 
would be more effective in proportion as the particle was smaller, 
and this is in accordance with the apparent fact that the molecules 
of hydrogen, hydrocarbon gas, and metallic vapors are sorted 
out, so to speak, to form the three different types of tails. 

But the experiments of Nichols and Hull in this country and 
of Lebedew in Russia, made independently in 1901, tend to 


confirm a long-standing surmise that it may be due to the Repulsion 
direct action of the waves of solar radiation upon extremely ^®^° J^^^"^ 

^ ^ action of 

small particles of matter. light-waves. 

Maxwell, years ago, showed that as a consequence of his electromagnetic 
theory of light (then new, but now almost universally accepted), a particle 
receiving light-rays ought to be repelled by a force the amount of which 
he computed. For particles of sensible magnitude the calculated force 
is insignificant as compared w-ith the solar attraction, but for particles 
— say, a hundred thousandth of an inch in diameter — it many times 
exceeds that attraction. Various unsuccessful attempts have been made 
to detect such a force experimentally, but at last the physicists seem to 
have overcome the difficulties, and their result practically agrees with 
Maxwell's prediction. This theory is supplementary to the electi'ical 
rather than contradictory, as the repulsive force of light is due to an 
electromagnetic reaction, and it is not unlikely that the particles repelled 
may carry electric charges. 

It has also been attempted to account for the repulsion by an indirect Evaporation 
action resulting from the heating of the surfaces of the almost infinitesi- theory, 
mal particles on the side next to the sun. 

There is no reason to suppose that the matter driven off to form 
the tail is ever recovered hy the comet. It probably remains in 
space, to be picked up by any large masses which the particles 
may meet. 

Whenever a comet comes near to the sun or to one of the 
larger planets, it is subjected to forces which tend to pull it 
to pieces, and, as the mutual attraction between its particles 
is extremely feeble, it sometimes happens that it is separated 
into several portions, as was the case with Biela's comet in 
1846, with the great comet of 1882, and with Brooks' comet 
of 1889. Indeed, it seems likely that all along its course it Disintegra- 
loses portions of its substance, so that at each successive return *^°^ °^ 


to perihelion it becomes smaller and finally ceases to exist 
as a recognizable " body," the scattered particles traveling by 
themselves until they fall upon some larger body as "shooting- 




features of 
the great 
comet of 


raphy of 

503. Unexplained and Anomalous Phenomena. — A curious 
phenomenon, not yet explained, is the dark stripe which in 
the case of a large comet nearing the sun runs down the 
center of the tail, looking very much as if it were a shadow 
of the comet's head. It is certainly not a shadow, however, 
because it usually makes more or less of an angle with the 
sun's direction. It is well shown in Fig. 162. When the 
comet is at a greater distance from the sun this central stripe 
is usually bright, as in Fig. 165. Indeed many, perhaps most, 
small comets, instead of the usual hollow, horn-shaped tail, 
show only this narrow streak, smaller in diameter than the 
comet's head, — as if the material repelled by the sun fol- 
lowed around the coma and left it 
only at the point remotest from the 

Not infrequently, however, comets 
possess anomalous tails, — usually 
in addition to the normal tail, but 
sometimes substituted for it, — tails 
directed sometimes straight towards 
the sun and sometimes nearly at right angles to that direction. 
The great comet of 1882 also carried with it for a time a 
faintly luminous " sheath," which seemed to envelop the comet 
itself and that portion of the tail near the head, projecting 2° 
or 3° forward towards the sun. For some days, moreover, it 
Avas accompanied by little clouds of cometary matter, which 
left the main comet, like smoke puffs from a bursting bomb, 
and traveled off at an angle until they faded away. None of 
these appearances contradict the theory outlined above, but they 
cannot be said to be explained by it, — evidently we have not 
yet the whole story. 

504. Photography of Comets. — It is not unlikely that photog- 
raphy will give us light on the subject, for the sensitive plate 
reveals in the tail of the comet (not in the head) many interesting 

Fig. 165. — Bright-Centered Tail of 
Coggia's Comet, June, 1874 



details which are wholly invisible to the eye ; partly, it is likely, 
because of the cumulative action of the feeble light during a long 
photographic exposure, and partly, also, because the light of a 
comet's tail probably resembles that of the positive " brush " from 
a charged electrode in being very rich in ultra-violet rays, which 
act powerfully in photography, but do not affect the eye. 

The first photograph of a comet was obtained by Bond in 
1858, — only a partial success and but little known. The next 
was in 1881, when Henry Draper in New York and Huggins 
in England photographed Tebbutt's comet, and in 1882 the 
great comet was well photographed by Gill in South Africa. 
Fig. 166 is a series 
of photographs of 
Swift's comet of 
1892 by Barnard. 
The tail was barely 
visible to the 
naked eye, and the 
peculiar features 
exhibited in the 
photograph were 
not visible at all. 

Fig. 167 is from 
Hussey's beautiful 

photograph of Rordame's comet of 1893, for which we are 
indebted to the kindness of Professor Holden. 

Since in photographing a comet the camera is kept pointed at 
the head, which is moving more or less rapidly among the stars, 
the star images, during the long exposure, are drawn out into 
parallel streaks, as seen in the photograph. The little irregu- 
larities are due to faults of the driving clock and vibrations of 
the telescope and atmosphere. 

The knots and streamers, which in the photographs charac- 
terize the comet's tail, were none of them visible in the telescope 

Fig. 1(36. — Swift's Comet of 1892 

comet of 



Fig. 167. — Kordame's Comet, July 13, 1893 
From photograph by W. J. Hussey, at the Lick Observatory 



and differ from those shown upon plates preceding and follow- 
ing. Other plates of Rordame's comet, made on the same 
evening a few hours earlier and later, indicate that these knots 
were swiftly receding from the comet's head at a rate exceeding 
150000 miles an hour. 

Fig. 168 is a photograph (also by Barnard) of Gale's comet 
(May, 1894). It was moving through a crowd of stars. 

In three cases already mentioned (Sec. 479) comets have been 
discovered by photography. 

505. Danger from Comets. — 
We close the chapter with a few 
remarks upon a subject which 
has been much discussed. 

It has been supposed that 
comets might do us harm in 
two ways, — either by actually 
striking the earth or by falling 
into the sun, and thus pro- 
ducing such an increase of 
solar heat as to burn us up. 

As regards collision with a 
comet, there is no question that 
the event is possible. In fact, 

Fig. 168. — Gale's Comet, May 5, 1894 

if the earth lasts long enough, it is practically sure to happen ; 
for there are several comets whose orbits pass nearer to our 
own than the semidiameter of the comet's head, and at some 
time the earth and comet, if the comet lasts long enough, will 
certainly come together. 

As to the consequence of such a collision it is impossible to 
speak positively, for want of sure knowledge of the constitution 
of the comet. If the theory which has been presented is true, 
everything depends on the size of the separate particles which 
form the main portion of the comet's mass. If they weigh 
tons, the bombardment experienced by the earth when struck 


motion of 
knots in tail 
of comet. 


of collision 
with comet. 

that col- 
lision would 
be harmless 
to the earth. 


by a comet would be a very serious matter ; if, as seems mucli 
more likely, they are for the most part smaller than pinheads, 
the result would be simply a splendid shower of shooting-stars. 
In 1861 the earth actually passed unnoticed through the tail of 
the great comet of that year. 

Such encounters will, however, be very rare ; if we accept 
the estimate of Babinet, they ought to occur once in about 
15 000000 years in the long run. 

A danger of a different sort has been suggested, — that 
if a comet were to strike the earth, our atmosphere would 
be poisoned by the mixture with the gaseous components of 
the comet. Here, again, the probability is that on account 
of the low density of the cometary matter no sufficient amount 
would remain in the air to do any mischief at the earth's 
Possible fall 506. Effect of the Fall of a Comet into the Sun. — As to this, 
o come -^ ^^ 1^^ stated that, except in the case of Encke's comet, there 

into the Sim . *^ ^ r ' 

is no evidence of any action going on that would cause a now 
existing periodic comet to strike the sun's surface ; it is, how- 
ever, doubtless possible, perhaps not improbable, that a comet 
may sometime enter the system from without, so accurately 
aimed as to hit the sun. 

But in that case it is not likely that the least mischief would 

be done. If a comet with a mass equal to y o'oV'o'o ^^ ^^® earth's 

mass were to strike the sun's surface with the parabolic velocity 

Probably no of nearly 400 miles a second, the energy of impact converted 

harm except ^^^^^ j^^^^^ would P^cnerate about as many calories of heat as the 

to the comet. ... . . . 

sun radiates in eight or nine hours. If this were all instantly 
effective in producing increased radiation at the sun's surface 
(increasing it, say, eightfold, for even a single hour), harm 
would doubtless follow; but it is practically certain that 
nothing of the sort would happen. The cometary particles 
would pierce the photosphere and liberate their heat mostly 
helotv the sola?- surface^ simply expanding, by some slight 


amount, the sun's diameter, and so adding to its store of 
potential energy about as much as it ordinarily expends in a 
few hours and postponing, by so much, the date of its final 
solidification. There might, and very likely would, be a flash 
of some kind at the solar surface as the shower of cometary 
particles struck it, but probably notliing that tlie astronomer 
would not take delight in watcliing. 


1. What would be the mean density, compared with air, of the spherical 
head of a comet 100000 miles in diameter and having a mass tooWo that 
of the earth, assuming the density of the earth to be 5.53 times that of 
water and the density of water 773 times that of air? Ans. About 49^00- 

2. What would be the diameter of such a comet if compressed to a 
density the same as that of the earth? ^j;^^, 171 miles. 

3. Can the dimensions of a comet's tail be determined with much 
accuracy ? If not, why not ? 

4. How can it happen that comets whose orbits nearly coincide within 
a distance of 100 000000 miles from the sun may have periods differing by 
hundreds of years? For example, the comets of 1880 and 1882, of which 
the first has a computed period of only 33 years, and the other of more 
than 600. 

5. In the case of two cometary orbits very nearly parabolic, and having 
the same very small perihelion distance, how would the ratio of their major 
axes be affected by a small difference in their perihelion velocities ? (See 
Sec. 320, remembering that, as the orbits are nearly parabolic, V^ must be 
very nearly equal to U^ when the comets pass perihelion.) 

6. If the repulsive force of the sun upon a particle of a comet's tail 
were just equal to the gravitational attraction (Sec. 502), what would be 
the path of that particle ? j^„g^ A straight line. 

7. If the repulsive force exceeded the gravitational attraction, what 
would be the nature of the path? 

Ans. An orbit convex toward the sun, hyperbolic if the repulsion 
varied inversely as the square of the distance, the sun being in the 
focus outside the cm-ve, i.e., at F" in Fig. 119, Sec. 314. 



8. What would be the path if the repulsive force were only very small 
as compared with the gravitational attraction ? 

Ans. An orbit of slightly greater major axis and period than that 
of the comet itself. 

9. Will a given comet (say Encke's) have precisely the same orbit on 
successive returns ? 

10. Why can we not infer with certainty that two comets which have 
orbits practically identical are themselves identical ? 

11. Can we, from spectroscopic observations of a comet, infer the rela- 
tive proportions of the luminous and non-luminous substances present in 
the comet? 

12. Is it probable that a comet can continue permanently in the solar 
system as a comet ? If not, why not, and what will become of it ? 

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Potsda,m Astrophysical Observatory 


Aerolites : their Fall and Physical Characteristics ; Cause of Light and Heat ; Prob- 
able Origin — Shooting-Stars : their Number, Velocity, and Length of Path — 
Meteoric Showers : the Radiant ; Connection between Comets and Meteors 


507. Meteorites, or Aerolites. — Occasionally bodies fall upon 
the earth out of the sky, coming to us from outer space. Until they 
reach our air they are invisible, but as soon as they enter it they 
blaze out, become conspicuous, and the pieces which fall from 
them are called meteorites^ aerolites^ or simply meteoric stones. 

If the fall occurs at night, a ball of fire is seen, which moves Circum- 
with an apparent velocity depending upon the distance of the ^^'^^^^^^ ^^ 
meteor and the direction of its motion, and is generally followed aerolites. 
by a luminous train, which sometimes remains visible for many 
minutes after the meteor itself has disappeared. The motion 
is usually somewhat irregular, and here and there along its 
path the fire-ball throws off sparks and fragments and changes 
its course more or less abruptl}^ Sometimes it vanishes by simply 
fading out in the distance, sometimes by bursting like a rocket. 

If the observer is near enough, the flight is accompanied by 
a heavy continuous roar, like that of a passing railway train, 
accentuated now and then by violent detonations ; the noise is 
frequently heard 50 miles away, especially the final explosion. Delay of 
The observer, however, must not expect to hear the explosion s*^^^"^^ ^f 

. explosion. 

when he sees it. Sound travels only about 12 miles a minute, 
so that there is often an interval of several minutes between 
the visible bursting and its report. 

Size of 




Number of 
since 1800. 


of which the 
fall was 


If the fall occurs by day, the luminous appearances are 
mainly wanting, though sometimes a white cloud is seen, and 
even the train may be visible. In a few cases, aerolites have 
fallen almost silently, and without warning. 

508. The Aerolites themselves. — The mass that falls is some- 
times a single piece, but more usually there are many fragments, 
sometimes to be counted by thousands. At the Pultusk " fall," 
in 1869, the number was estimated to exceed 100000, mostly 
very small. The pieces weigh from 500 pounds to a few grains, 
the aggregate mass occasionally amounting to more than a ton. 
The largest single mass, so far as known, is one that fell at 
Knyahinya in 1866, weighing 647 pounds. 

By far the greater number of aerolites are stones, but a few 
— one or two per cent of the whole number — are pieces of 
nearly pure iron more or less alloyed with nickel. 

The total number of meteorites which have fallen and been 
gathered into our cabinets since 1800 is about 275. The only 
instances in which purely iron meteorites have been actually 
seen to fall and are represented by specimens in our cabinets 
are the eight following, viz. : 

Agram, Croatia, Austria 1751 

Dickson County, Tennessee, U.S 1835 

Braunau, Bohemia 1847 

Victoria West, South Africa 1862 

ISTedagollah, Arabia 1870 

Rowton, England 1876 

Mazapil, Mexico . 1885 

Cabin Creek, Arkansas, U.S 1886 

There are about as many more which contain large quantities 
of iron and by some authorities have been reckoned as "irons" ; 
nearly all meteorites contain a large percentage of the metal, 
either in the metallic form or as sulphid. 

About 80 of the 275 fell within the United States, the most 
remarkable being those of Weston, Conn., in 1807; New 

ber ill 


Concord, Ohio, in 1860; Anmna, Iowa, 1875 ; Emmet Comity, 
Iowa, 1879 (largely iron) ; and Cabin Creek, Ark., 1886. 

Our cabinets at present contain specimens of somewhat more Total num- 
than three hundred meteors which have been seen to fall, besides 
a nearly equal number of other bodies, — mostly masses of iron 
which, from the circumstances of their finding and the peculiari- 
ties of their constitution, are supposed to be of meteoric origin. 

The finest collection in the world is that at Vienna. The collection of 
the British Museum and that at Paris are also noteworthy ; and in this 
country the cabinet of Yale University is especially rich. 

509. Appearance and Constitution of the Meteorites. — The Appearance 
most characteristic external feature of an aerolite is the thin ^^ ^^^^ eoi- 

ites: the 

black crust which covers it, usually, but not always, glossy crust. 
like varnish. It is formed by the fusion of the surface in the 
meteor's swift motion through the air, and in some cases pene- 
trates deeply into the mass through veins and fissures. It is 
largely composed of oxid of iron and is almost always strongly 
magnetic. The crusted surface usually exhibits pits and hol- 
lows, called "thumb-marks" because they look like prints pro- 
duced by thrusting the thumb into a piece of putty. These 
cavities are explained by the burning out of certain more fusible 
substances during the meteor's flight. 

On breaking, the stone is sometimes found to be compara- internal 
tively fine grained, but usually is made up of crystalline lumps s*^"^^<^^"^'®- 
and globules, and sometimes has a considerable portion of solid 
iron scattered throughout the mass in grains as large as a pin- 
head or bird shot. 

Twenty-seven of the chemical elements, including argon and Chemical 
helium, have been found in meteorites, but not a single neiv ^i^'"^"^^- 

^ peculiar 

element. Many of the minerals of which the meteorites are minerals. 
composed present a great resemblance to terrestrial minerals of 
volcanic origin, but there are also many which are peculiar and 
not found on the earth. 



Path and 

The occasional presence of carbon is to be especially noted ; 
and in a meteor which fell in Russia in 1887, the carbon 
appeared to be in a crystalline form, identical with the black 
diamond, though in particles exceedingly minute. 

Fig. 169 is from a photograph of a fragment of one of the meteoric 
stones which fell at Gross Divina, Hungary, in 1837; weight about twenty- 
four pounds. 

510. Path and Motion. - — When a meteor has been well 
observed from a number of different stations a considerable 

when first 

Length of 


Fig. 169. — The Gross Divina Meteorite 

distance apart, its path with reference to the surface of the 
earth can be computed. 

It is found that it usually first appears at an altitude of 
about 80 or 100 miles and disappears at a height of from 
5 to 10. The length of the path is generally between 50 
and 500 miles, though in some cases it has been much greater. 
In 1860 one passed from over Lake Michigan across the 
country and fell into the sea beyond Cape May; and in 1876 
a great meteor traversed the country from Kansas to northern 


The velocity ranges from 10 to 40 miles a second in the Velocity, 
earlier part of its course, but is very rapidly and greatly 
reduced by the resistance of the atmosphere, so that when the 
surface of the earth is reached it is often not more than 400 or 
500 feet a second. In one case (a meteor that fell near Upsala, 
Sweden, in January, 1869) several of the stones struck upon 
the ice of a lake and rebounded without breaking the ice or 
damaging themselves. 

The average velocity with which these bodies enter the air Meteorites 
seems to be very near the parabolic velocity of 26 miles a ^^® visitors 

'^ ^ _ "^ ^ _ from distant 

second, due to the sun's attraction at the earth's distance, — just regions, 
as should be the case, if, like the comets, they come to us from 
distant regions of space. 

511. Observation of Meteors. — The object of the observer observation 
should be to obtain as accurate an estimate as possible of the o^^Jieteors. 
altitude and azimuth of the meteor at moments which can be 
identified, and also of the time occupied in traversing definite 
portions of the path. 

By night the stars furnish the best reference points from Determi- 
which to determine its position. By day one must take advan- ^^^lon of 
tage of natural objects and buildings to define the meteor's altitude and 
place, the observer marking the precise spot where he stood azimuth, 
when the meteor disappeared behind a chimney, for instance, or 
was seen to burst just over a certain branch in a tree. By taking 
a surveyor's instrument to the place afterwards it is then easy 
to translate such data into altitude and hearing. 

As to the time of flight, which is required in order to deter- Time of 
mine the meteor's velocity, it is usual for the observer to begin ^^s^^^- 
to repeat rapidly some familiar verse of doggerel when the 
meteor is first seen, reiterating it until the meteor disappears. 
Then, by rehearsing the same before a clock, the number of 
seconds can be pretty accurately determined. 

512. Explanation of Heat and Light. — These are simply due 
to the intense condensation of the air before the swiftly moving 



to condensa 
tion of air 
in front of 

meteor, and consequent destruction of the meteor's energy. 
Heating due The resistance, due to condensation, amounts in many cases to 
the back-pressure of hundreds of pounds upon a square inch; 
and most of the energy of the meteor destroyed in this way is 
transformed into heat, largely imparted to the air, but to a 
considerable extent expended upon the surface of the meteor, 
fusing it and producing the crust. 

If a moving body whose mass is M kilograms, and its velocity 
V kilometers per second, is stopped by a resistance, its energy is 
almost entirely converted into heat, and the number of calories 
(Sec. 267) developed is given (approximately) by the equation 

Formula for 
amount of 
heat de- 

Virtual tem- 
perature of 
air en- 
is that of 
a blowpipe 

In bringing to rest a body having a mass of one kilogram and 
a velocity of forty-two kilometers, or 26 miles a second, the quan- 
tity of heat developed is enormous, — nearly 212000 calories, — 
vastly more than sufhcient to fuse it, even if it were made of the 
most refractory material. As Lord Kelvin has shown, the ther- 
mal effect of the rush through the air is the same as if the meteor 
were immersed in a blowpipe flame having a temperature of 
many thousand degrees ; and it is to be noted that this tempera- 
ture is independent of the density of the air through which the 
meteor is passing. The quantity of heat developed in a given 
time is greater, of course, where the air is dense, but the tempera- 
ture produced in the air itself at the surface where it encounters 
the moving body is the same whether it be dense or rare. 

This rise of temperature is due to the fact that the gaseous 
molecules strike the surface of the meteor as if the meteor were 
at rest and the molecules themselves Avere moving with speed 
correspondingly increased. (According to the kinetic theory 
of gases, the " temperature " of a gas depends entirely upon 
the mean velocity square of its molecules.) 

When the moving body has a velocity of one and one-half 
kilometers per second the virtual temperature of the surrounding 


air is about that of red heat. When the velocity reaches 

thirty kilometers per second the amount of heat developed is 

15^, or 225, times as great, and the surface is acted upon as if 

the surrounding gas were a blowpipe flame, as has been said ; Formation 

the surface of the meteor is fused and the liquefied portion is ^^ ^'^'^^^^' 

aud train. 

continually swept off by the rush of air, condensing as it cools 
into the luminous dust that forms the train. The fused surface 
is continually renewed until tlie velocity falls below two kilo- 
meters a second, or thereabouts, when it solidifies and forms 
the crust. 

As a general rule, therefore, tiie fragments are hot if found Stones some- 
soon after their fall ; but if the stone is a laro-e one and falls ""^^ !:" ^ , 

' ^ when found. 

nearly vertically, so as to have a short path through the air, the 
heating effect will be confined to its surface, and, owing to the 
low conducting power of stone, the center may still remain 
intensely cold for some time, retaining nearly the temperature 
which it had in interplanetary space. It is recorded that one 
of the fragments of the Dhurmsala, India, meteorite, which fell 
in 1860, Avas found in moist earth half an hour or so after the 
fall coated with ice. 

One unexplained feature of the meteoric trains deserves Unexplained 
notice. They often remain luminous for a long- time, some- P^osphores- 

"^ ^ " cence of 

times as much as half an hour, and are carried by the wind trains, 
like clouds. It is impossible to suppose that such a cloud of 
impalpable dust remains white-hot for so long a time in the 
cold upper regions of the atmosphere, and the question of its 
enduring luminosity or phosphorescence is an interesting and 
a puzzling one. 

513. The Origin of Meteors. — The high velocity with which 
many enter our atmosphere makes it quite certain that they at 
least had not a terrestrial origin. A body projected from the 
earth could never return with higher velocity than that of pro- 
jection, and any velocity exceeding about 7 miles a second (the 
parabolic velocity at the earth's surface — Sec. 319) would carry 



come to us 
as astronom- 
ical bodies. 

can be 

Theories of 
their origin. 

Iron meteor- 
ites perhaps 
of stellar 

results : 
some meteor- 
ites perhaps 

the body permanently away from the earth, never to return 
unless after many revolutions around the sun. Most meteors, 
if not all, come to us as astronomical bodies, moving like planets 
or comets ; as to their origin, we can only speculate. 

At the same time we find in our cabinets many distinct 
classes of them, and in each class all the meteors which com- 
pose it resemble each other so closely as to suggest the idea 
that they must have had a common source, or at one time 
formed portions of a single mass ; but where and when ? 

Some have maintained that they were projected from lunar 
volcanoes, ages ago perhaps (for lunar volcanoes are now inac- 
tive), and that since that time they have been moving around 
the sun like planets, until now encountered by the earth. 
Others refer them to similar imagined volcanic eruptions from 
the earth in some past age, and others consider them as pro- 
ceeding from the disintegration of comets. 

As to the iron meteorites, some believe that they have been 
ejected from the sun or from a star, basing the opinion upon the 
remarkable fact that these meteoric irons are usually "soaked 
full" of occluded gases, — hydrogen, helium, and carbon oxids, 
which can be extracted from them by well-known methods. It is 
argued that the iron could have absorbed these gases only when 
immersed in a hot dense atmosphere saturated with them, — a 
condition existing, so far as known, only on the sun and stars. 

An investigation by the late Professor Newton, however, 
shows that about ninety per cent of the aerolites, for the deter- 
mination of whose orbits we have sufficient data, were moving 
around the sun before their encounter with the earth in paths 
not parabolic, but resembling those of the short-period comets, 
or more eccentric asteroids, and nearly all direct, suggesting a 
planetary rather than a stellar origin; they might possibly be 
minute outriders of the asteroid family. 

Lord Kelvin suggested many years ago that meteors may 
have acted in conveying germs of life from one part of the 


universe to another, — a suggestion, however, not generally Theory of 
accepted, since they seem to have passed through conditions ^^^teors as 

^ ^ «/ ^ o carriers of 

of temperature which must have destroyed all life. life im- 

514. Number of the Aerolites. — As to the number of these P^o^^bie. 
bodies which strike the earth, it is difficult to make a trustworthy 
estimate. We generally add to our cabinets each year speci- 
mens of from two to six meteors which have been seen to fall. 

But for one that is found, even of the meteors whose flight has Number of 
been observed, a dozen are missed ; and if we include all that aerolites, 
were not seen, or that fell unobserved on the ocean or in regions 
from which no report could come, the sum total must be very 
great. Schreibers estimated the number at seven hundred a 
year. Reichenbach puts it at three or four thousand, but this 
is probably excessive. 


515. Their Nature and Appearance. — These are the swift- shootmg- 
moving, evanescent, starlike points of light which may be seen ^^ars: 
every few moments on any clear moonless night. They make minute aero- 
no sound, nor (perhaps with one exception, to be noted later) ^^^^s, but 
has anything been known to reach the earth's surface from clouds. 
them, not even in the greatest " meteoric showers." 

For this reason it may be well to retain provisionally the old 
distinction between them and the large meteors from which aero- 
lites fall. It is quite probable that the distinction has no real 
ground, that shooting-stars are just like other meteors, except 
in size, being so small that they are entirely consumed in the 
air; but then, on the other hand, there are some things which 
favor the idea that the two classes of bodies differ in constitu- 
tion about as asteroids do from comets. 

516. Number of Shooting-Stars. — Their number is enormous. 
A single ordinary observer averages from four to eight an hour ; 
one used to observation, well situated and on a moonless night. 



Their num- 
ber ten to 
twenty mil- 
lion daily. 

hourly num- 
ber in morn- 
ing twice as 
great as in 

tion of 
path, etc. 

will see at least twice as many; Schmidt of Athens sets the 
average number at fourteen. If the observers are sufficiently 
numerous and so organized as to be sure of noting all that are 
visible from their station, about eight times as many will be 

On this basis Professor Newton has estimated that the total 
number which enter our atmosphere daily must be between 
ten and twenty million^ the average distance between them 
being over 200 miles; and besides those which are visible to 
the naked eye there is an immensely larger number so small as 
to be observable only with the telescope. Dr. See estimates 
the number of these as at least one hundred million daily. 

The average hourly number about six o'clock in the morning 
is double the hourly number in the evening, and the meteors 
move much swifter, the reason being that in the morning we 
are on the front of the earth as regards its orbital motion, while 
in the evening we are in the rear. (The earth's orbital motion 
is always directed towards a point on the ecliptic about 90° west 
of the sun.) In the evening, therefore, we see only such as 
overtake us. In the morning we see all that we either meet or 
overtake. This proportion of morning and evening meteors 
is precisely what it should be if they come to us indiscrimi- 
nately from all directions and with the parabolic velocity of 
26 miles a second. 

517. Elevation, Path, and Velocity. — By observations made 
at stations 30 or 40 miles apart (best by photography) it is easy 
to determine these data with some accuracy whenever meteors 
identifiable at the two or more stations make their appearance. 
It is found that on the average the shooting-stars appear at 
a height of about 74 miles and disappear at an elevation of 
about 50 miles, after traversing a course of 40 or 50 miles, with 
a velocity of from 10 to 30 miles a second, — about 25 on the 
average. They do not begin to be visible at so great a height 
as the aerolitic meteors, and they are more quickly consumed 


and therefore do not penetrate our atmosphere to so great a 
depth, — fortunately for us. 

518. Brightness, Material, etc. —Now and then a shooting- Brightness, 
star rivals Jupiter or even Venus in brightness. A consider- 
able number are like first-magnitude stars, but the great majority 

are faint. The bright ones generally leave trains, which some- Trains, 
times endure from five to ten minutes and then fold up and 
are wafted away by the air currents, which at 40 miles above 
the earth's surface ordinarily have velocities of from 50 to 75 
miles an hour. 

The swift meteors are usually of green or bluish tinge, while Color, 
those that move slowly are generally red or yellow. 

Occasionally it has been possible to get a '' snap shot," so to 
speak, at the spectrum of a meteor, and in it the bright lines Spectrum, 
of sodium and (probably) magnesium are fairly conspicuous 
among many others which cannot be identified by a hasty glance. 

Since these bodies are consumed in the air, all we can hope 
to get of their material is their ashes. In most places its collec- Meteoric 
tion and identification is hopeless; but Nordenskiold thought *^^®S' 
that it might be found in the polar snows. In Spitzbergen he 
therefore melted several tons of snow, and on filtering the water 
he actually detected in it a sediment containing minute globules 
of oxid and sulphid of iron. Similar globules have also been 
found in the products of deep-sea dredging. They may be 
meteoric; but what we now know of the distance to which smoke 
and fine volcanic dust is carried by the wind makes it not 
impossible that they may be of purely terrestrial origin. 

519. Probable Mass of Shooting-Stars. — We have no way Probable 
of determining the exact mass of such a body ; but from the "^^^^ ^^" 
light it emits, as seen from a known distance, an estimate can small. 
be formed not likely to be widely erroneous. 

A good ordinary incandescent lamp consumes about 150 foot- 
pounds of energy per minute for each candle-power. Assum- 
ing for the moment that the ratio of the total light emitted 



Data for 
of mass. 

of the mass 
of a shoot- 

mass proba- 
bly only a 
fraction of 
an ounce. 

{luminous energy) to the total energy consumed is the same for 
a meteor as for an electric lamp, we can compute the total 
energy of a meteor which shines with known brightness for a 
given time at a known distance. 

Suppose, for instance, that the shooting-star is at an average 
distance during its flight of 30 miles from the observer and 
appears as bright as a 16 candle-power lamp i of a mile away, 
and shines for five seconds (^^2 minute). The total luminous 
energy then equals 

150 X 16 X J 



^ j = 2 880000 foot-pounds. 

Suppose its velocity, F, to be 20 miles, or 105600 feet, per 
second. For the energy, E^ in foot-pounds, of a moving body 
whose velocity is V feet per second and its mass in pounds M, 
we have 

E =}y = —^rr- (nearly), whence, iM = 64 x — r,- 

Finally, then, in the case before us, 

,^ ., 2 880000 1 , ... . , 

M= 64 x -iAc^AA2 = oA po^i^ds, or 115 grams (nearly). 

This represents fairly the observed conditions for a very 
bright shooting-star. 

If a meteor converted all its energy into light, — ^.e., if its 
luminous efficiency were higher than that of a lamp, — this 
would give the mass much too great. On the other hand, if 
the meteor were only feebly luminous, the mass thus determined 
would be much too small. 

It seems likely that an average meteor and a good electric 
lamp do not differ widely in their luminous efficiency, and 
on this basis observations indicate that ordinary shooting-stars 
weigh only a fraction of an ounce, — from a grain or two up to 
100 or 150 grains. Some authorities, however, estimate the 
mass considerably higher. It all turns on the assumed "lumi- 
nous efficiency " of the shooting-stars. 


520. Effects produced by Meteors and Shooting-Stars. — (1) p:ffectsdue 
Meteors add contmually to the mass of the earth. If we assume " ^ ^! 

^ '^ meteoric 

20 000000 a day, each weighing J^ of a pound, the total amount matter 
would be about 50000 tons a year; and if the specific gravity P^'^cticaiiy 

•^ ^ . . insensible. 

of the meteoric dust averages the same as that of granite, it 
would take about eight hundred million years for the deposi- 
tion of a layer 1 inch thick on the earth's surface. 

(2) They diminish the length of the year in three ways : (a) by 
acting as a resisting medium, and so really shortening the major 
axis of the earth's orbit (just as the orbit of Encke's comet is 
shortened) ; (h) by increasing the mass of the earth and sun, and 
so increasing the attraction between them; (c) by increasing 
the size of the earth, thus slackening its rotation, lengthening 
the day, and so making fewer days in the year. 

Calculation shows, however, that on the preceding assump- 
tion as to the mass of the meteors, the combined effect would 
hardly amount to more than j-^-qq of a second in a million years. 

(3) ^ach meteor brings to the earth a certain amount of heat, 
developed in the destruction of its motion. According to the 
best estimates, however, all the meteors that fall upon the earth 
in a year supply no more heat than the sun does in about one 
tenth of a second. 

(4) They must necessarily render space imperfectly transparent 
if they pervade it throughout in any such numbers as in the 
domain of the solar system; but this effect, though doubtless 
real, is also so small as at present to defy calculation. 


521. There are occasions when the shooting-stars, instead of Meteoric 
appearing here and there in the sky at intervals of several ^^^°^^'^^'^- 
minutes, appear in showers of thousands; at such times they 
do not move at random, but all their paths diverge or radiate 
from a single point in the sky, known as the Radiant; i.e., The radiant. 



ture of 

their paths produced backward all pass through or near that 
point, though they do not usually start there. Meteors which 
appear near the radiant are apparently stationary, or describe 
paths which are very short, while those in the more distant 
regions of the sky pursue longer courses. 

The radiant keeps its place among the stars sensibly unchanged 
during the whole continuance of the shower, — for hours or days, 
it may be, — and the shower is named according to the place of 

The radiant 
an effect of 

Fig. 170. — The Meteoric Kadiant in Leo, Nov. 13, 1866 

the radiant among the constellations. Thus, we have the Leo- 
nids, or meteors whose radiant is in the constellation of Leo, 
the Andromedes (or Bielids), the Perseids, the Lyrids, etc. 

Fig. 170 represents the tracks of a large number of the Leonids of 1866, 
showing the position of the radiant near ^ Leonis. It show^s also the 
tracks of four meteors observed during the same time, which did not 
belong to the shower. 

The radiant is a mere effect of perspective. The meteors are 
all moving in lines nearly parallel when encountered by the 


earth, and the radiant is simply the perspective vanishing 2)oint 
of this system of parallels ; their paths all appear to converge, 
like the rails of a railway track for an observer looking upon 
it from a bridge. The position of the radiant on the celestial 
sphere depends entirely upon the direction of the motion of the 
meteors relative to the observer. For various reasons, however, 
the paths of the meteors, on account of irregularities in their 
form and surfaces, are not exactly parallel or straight, and in 
consequence the radiant is not a mathematical point, but a spot 
or patch in the sky, often covering an area of 3° or 4°. 

Probably the most remarkable of all the meteoric showers 
that have ever occurred was that of the Leonids, on November 
12, 1833. The number at some stations was estimated as high The Leonid 
as 200000 an hour for five or six hours. " The sky was as full ^^^^wer of 


of them as it ever is of snowflakes in a storm " and, as an old 
lady described it, looked "like a gigantic umbrella." 

522. Dates of Meteoric Showers. — Meteoric showers are evi- Fixed dates 
dently caused by the earth's encounter with a swarm of the ^* meteoric 

*^ ^ ^ ^ ^ showers. 

little bodies, and since this swarm or flock pursues a regular 
orbit around the sun, the earth can meet it only when she is 
at the point where her orbit cuts the path of the meteors ; this, 
of course, must always happen at or near the same time of the 
year, except as in the process of time the meteoric orbits shift 
their positions on account of perturbations. The Leonid 
showers, therefore, appear about November 15, and the Andro- 
medes about the 24th ; but both dates are slowly changing, the 
Leonids coming gradually earlier and the Andromedes later. 
Since 1800 the former have shifted from November 12 to the 
15th, and the latter from the 28th to the 24th since 1872. 

In some cases the meteors are distributed along their whole Annual re- 
orbit, forminsf a sort of ring: and rather widely scattered. In ^^™^^®.*jf 

o o J thePerseids. 

that case the shower recurs every year and may continue for 
several weeks, as is the case with the Perseids, or August 
meteors. On the other hand, the flock may be concentrated, 



with the 
Leonids and 

istics of 

and then a notable shower will occur only on the day when the 
earth and the meteors arrive together at the orbit crossing. 
This is the case with both the Leonids and the Andromedes, 
though the latter are already getting pretty widely scattered. 
The showers then occur, not every year, but only at intervals 
of several years, and always on or near the same time of the 
month. For the Leonids the interval is about thirty-three 
years, and for the Andromedes usually thirteen, but sometimes 
only six or seven. 

The meteors which belong to the same group have certain 
family resemblances. The Perseids are yellow and move with 
medium velocity. The Leonids are very swift (we meet them), 
and they are of a bluish green tint, with vivid trains. The 
Andromedes are sluggish (they overtake the earth), are reddish, 
being less intensely heated than the others, and usually have 
only feeble trains. 

now recog- 

About one hundred meteoric radiants are now recognized and cata- 
logued. The most conspicuous of them, except those already named, 
are the following: the Draconids, January 2; Lyrids, April 20; Aqua- 
riids I, May 6; Aquariids II, July 28; Orionids, October 20; Geminids, 
December 10. 


523. Stationary Radiants. — When a meteoric shower per- 
sists for days and even weeks, as do the Perseids for instance, 
the radiant, as a rule, gradually shifts its position among the 
stars, on account of the change in the direction of the earth's 
motion, — as it ought to, since the place of the radiant depends 
upon the combination of the earth's motion with that of the 

Mr. Denning of Bristol (England), for many years an assid- 
uous observer of meteors, claims, however, to have discovered 
numerous cases in which the radiant of a long-continued shower 
remains absolutely stationary/ ; and he presents as typical the 
Orionids, which scatter along from about October 10 to 24, all 


the time, according to his observations, keeping their radiant 
close to the star v Orionis. 

No satisfactory explanation of such fixity of the radiant yet Difficult to 
appears, though certain mathematical investigations by Turner ^^pi^m. 
of Oxford (on the disturbing effect of the earth upon meteors 
passing near her) look promising and may resolve the problem; 
but some high authorities still remain skeptical as to the fact. 

524. The Mazapil Meteorite. — As has been said, during these Meteorite 
showers no sound is heard, no sensible heat perceived, nor have ^ ^^? ^ ® 

^ _ during 

any masses ever reached the ground; with the one exception, shower of 
however, that on Nov. 27, 1885, a piece of meteoric iron fell at ^^eiids. 
Mazapil, in northern Mexico, during the shower of Andromedes, 
or " Bielids," which occurred that evening. 

Whether the coincidence was accidental or not, it is inter- 
esting. Many high authorities speak confidently of this piece 
of iron as being a piece of Biela's comet itself. 

This brings us to one of the most remarkable discoveries of 
nineteenth-century astronomy. 


525. At the time of the great meteoric shower of 1833, Pro- Olmsted's 
fessors Olmsted and Twining- of New Haven were the first to ^'^cog^iition 

^ ^ ... . ^^ meteoric 

recognize the radiant and to point out its significance as indi- swarms as 
eating the existence of a swarm of meteors revolving around cometiike. 
the sun in a permanent orbit; Olmsted even went so far as to 
call the body a " comet." Others soon showed that, in some 
cases at least (Perseids), the meteors must be distributed in a 
complete ring around the sun, and Erman of Berlin developed 
a method of computing the meteoric orbit when its radiant is 

In 1864 Professor Newton of New Haven showed by 
an examination of the old records that there had been a 
number of great meteoric showers in November, at intervals of 



of shower 
of 1866. 

Failure in 

shower of 

thirty-three or thirty-four years, and he predicted confidently a 
repetition of the shower on Nov. 13 or 14, 1866. The shower 
occurred as predicted and was observed in Europe ; and it was 
followed by another in 1867, which was visible in America, the 
meteoric swarm being extended in so long a procession as to 
require more than two years to cross the earth's orbit. Neither 
of these showers, however, was equal to the shower of 1833. 
The researches of Newton, supplemented by those of Adams, 
the discoverer of Neptune, showed that .the swarm moves in a 
long ellipse with a thirty-three-year period. 

A return of the shower was expected in 1899 or 1900, but 
failed to appear, though on Nov. 14-15, 1898, a considerable 
number of meteors were seen, and in the early morning of 
Nov. 14-15, 1901, a well-marked shower occurred, visible over 
the whole extent of the United States, but best seen west of the 
Mississippi, and especially on the Pacific coast. At a number 
of stations several hundred Leonids Avere observed by eye or by 
photography, and the total number that fell must be estimated 
by tens of thousands. The display, however, seems to have 
nowhere rivaled the showers of 1866-67, and these were not to 
be compared with that of 1833. . 

It is not impossible that another minor shower may be 
observed in 1902. 

Cause of 
failure in 

The calculations of Downing and Stoney show that the failure in 1900 
was probably due to perturbations of the meteors by the action of Jupiter, 
Saturn, and Uranus during their absence from the neighborhood of the 
sun, causing the main body to pass at a distance of nearly 2 000000 miles 
below the orbit of the earth. 

Schiapa- 526. Identification of Meteoric Orbits with Cometary. — The 

reih'sidenti- j-gggarches of Newtoii and Adams had awakened lively interest 

ncation of *^ 

orbit of in the subject, and Schiaparelli, a few weeks after the Leonid 

Perseids. showcr, published a paper upon the Perseids, or August meteors, 

in which he brought out the remarkable fact that they are 



moving in the same orbit as that of tlie bright comet of 1862, 
known as Tuttle's comet. Shortly after this Leverrier published 
his orbit of the Leonid meteors, derived from the observed posi- 
tion of the radiant in connection with the periodic time assigned Leverrier 
bv Adams ; and almost simultaneously, but without any idea ^"f ^P" 

'^ ... polzer on 

of a connection between them, Oppolzer published his orbit of orbit of 


Fig. 171. — Orbits of Meteoric Swarms 

Tempel's comet of 1866, and the two orbits were at once seen 
to be practically identical. Now a single coincidence might be 
accidental, but hardly two. 

Five years later came the shower of the Andromedes, follow- Andromedes 
inP" in the track of Biela's comet, and among* more than one ^^^^ 

^ _ ^ . ^ ^ comet. 

hundred of the distinct meteor swarms now recognized Prof. 
Alexander Herschel finds five others which are similarly related 
each to its special comet. It is no longer possible to doubt 
that there is a real and close connection between these meteors 
and their attendants. Fig. 171 represents four of these cometo- 
meteoric orbits. 



Nature of 
comets and 
meteors not 
yet deter- 

tion of 
swarm into 
a ring. 

Eings older 
than com- 

The mete- 
oritic hy- 

527. Nature of the Connection. — This cannot be said to be 
ascertained. In the case of the Leonids and the Andromedes 
the meteoric swarm follows the comet, but this does not seem to 
be so in the case of the Perseids, which scatter along more or 
less abundantly every year. 

The prevailing belief at present seems, on the whole, to be 
that the comet itself is only the thickest part of a meteoric 
swarm, and that the clouds of meteors scattered along its path 
result from its disintegration. 

It is easy to show that if a comet really is such a swarm it is 
likely to break up gradually more and more at each return to 
perihelion, and at every near approach to one of the larger 
planets, dispersing its constituent particles along its path until 
the compact swarm has become a diffuse ring. The different 
parts of the comet are at different distances from the sun, and 
there is almost no sensible mutual attraction between them, 
the mass is so minute. The attraction of the sun or planet is 
therefore likely to cause the separation that has been referred to. 

The longer the comet has been moving around the sun, the 
more uniformly the particles will be distributed. The Perseids 
are supposed, therefore, to have been in the system for a long 
time, while the Leonids and Andromedes are believed to be 
comparatively new-comers. Leverrier, indeed, has gone so far 
as to indicate the year a.d. 126 as the time at which Uranus 
captured Tempel's comet and brought it into the system (as 
illustrated by Fig. l72). But the theory that meteoric swarms 
are the product of cometary disintegration assumes that comets 
are compact aggregations when they enter the system, which is 
by no means certain. 

528. Sir Norman Lockyer's Meteoritic Hypothesis. — Within 
the last twenty years Sir Norman Lockyer has been enlarging 
greatly the astronomical importance of meteors. The probable 
meteoric constitution of the zodiacal light, as well as of Saturn's 
rings, and of the comets, has long been recognized ; but he goes 



much further and maintains that all the heavenly bodies are 
either meteoric swarms, more or less condensed, or the final 
products of such condensation. Upon this hypothesis he 
attempts to explain the evolution of the planetary system, the 
phenomena of temporary and variable stars, the various classes 

Fig. 172. 

Origin of the Leonids 

of stellar spectra, the forms and structure of the nebulce, — in 
fact, pretty much everything in the heavens from the aurora 
borealis to the sun. As a working hypothesis his theory is 
unquestionably suggestive and has attracted much attention, 
but it encounters serious difficulties in details and cannot be 
said to be as yet " accepted." 




1. If a compact swarm of meteors were now to enter the system and be 
deflected by the attraction of some planet into an elliptical orbit around 
the sun, would the swarm continue to be compact ? If not, what would be 
the ultimate distribution of the meteors ? 

2. What is the probable relative age of meteoric swarms and meteoric 
rings as members of the solar system ? 

3. Assuming that the earth encounters 20 000000 meteors every 24 
hours, what is the average number in a cubic space of 1000 000000 cubic 
miles (i.e., a cube 1000 miles on each edge)? Ans. About 250. 

4. If space were occupied by meteors uniformly distributed 100 miles 
apart on three sets of lines perpendicular to each other, how many would 
be encountered by the earth in a day? Ans. 78 700000. 

Note. — In this cvibical arrangement the average distance between the meteors 
much exceeds 100 miles. If they were packed as closely as possible, consistently" 
with the condition that the distance between two neighbors should nowhere he less 
than 100 miles, the number would be increased by nearly forty per cent. 

Lick Observatory 


Their Nature, Number, and Designation — Star-Catalogues and Charts — The Photo- 
graphic Campaigns — Proper Motions, Radial Motions, and the Motion of the 
Sun in Space — Stellar Parallax 

529. Our solar system is an island in space, surrounded by The solar 
an imiTfense void inhabited only by meteors and comets. If system an 

•^ "^ , _ island in 

there were any body a hundredth part as large as the sun within space. 
a distance of a thousand astronomical units, its presence would 
be indicated by disturbances of Uranus and Neptune, even if it 
were itself invisible. 

The nearest star, so far as known at present, is at a distance Distance of 
of more than 275000 astronomical units, — so remote that, seen ^eaiest&ai, 
from it, our sun would look about like the pole-star, and no 
telescope ever yet constructed would be able to show a single 
one of all the planets of the solar system. 

That the stars are suns^ i.e.^ bodies of the same nature The stars 
as our own sun, composed largely of the same substances ^^ ^^^^\ 
and under similar physical conditions, is shown by their spec- greatly in 
tra. Each star has its incandescent photosphere surrounded ^^^® ^""^^ 

1 1 1 -T • 1 • intrinsic 

by a gaseous envelope, and while in a general way their brilliance, 
spectra resemble each other as human faces do, each has its 
own peculiarities ot detail. Small as they appear to us, they 
are many of them immensely larger and hotter than the sun; 
others, however, are smaller and cooler, and some hardly shine 
at all. They differ enormously among themselves in mass, 
bulk, and brightness, not being as much alike as individuals 
of a single race usually are, but differing as widely as whales 
from minnows. 




visible to 
the naked 

Total num- 
ber visible 
by tele- 

The con- 

530. Number of the Stars. — Those that are visible to the 
eye, though numerous, are by no means countless. If we 
examine a limited region, as, for instance, the bowl of " The 
Dipper," we shall find that the number we can see within it is 
not very large, — hardly a dozen, even on a very dark night. 

In the whole celestial sphere the number of stars bright 
enough to be distinctly seen by an average eye is between 
six and seven thousand, and that only in a perfectly clear 
and moonless sky; a little haze or moonlight will cut down 
the number by fully one half. At any one time not more than 
two thousand or twenty-five hundred are fairly visible, since, of 
course, one half are below the horizon and near it the small 
stars (which are vastly the most numerous) disappear. The 
total number which could be seen by the ancient astronomers 
well enough to he observable with their instruments is not quite 
eleven hundred. 

With even the smallest telescope the number is enormously 
increased. A common opera-glass brings out at least one hun- 
dred thousand, and with a 21^-inch telescope Argelander made 
his Durchmusterung of stars north of the equator, three hun- 
dred and twenty-four thousand in number. The Yerkes tel- 
escope, 40 inches in diameter, probably reaches over one 
hundred million. 

531. Constellations. — The stars are grouped in so-called 
"constellations," many of which are extremely ancient, all those 
of the zodiac and all those near the northern pole being of pre- 
historic origin. Their names are, for the most part, drawn from 
the Greek and Roman mythology, many of them being con- 
nected in some way or other with the Argonautic expedition. 

In some cases the eye, with the help of a lively imagination, 
can trace in the arrangement of the stars a vague resemblance 
to the object which gives name to the constellation, as in the 
case of Draco for instance, but generally no reason is obvious 
for either name or boundaries. 


Of the sixty-seven constellations now generally recognized, forty-eight Sixty-seven 
have come down from Ptolemy, the others having been formed since 1600 ^^ow recog- 
by later astronomers, in order to embrace stars not included in the old "^^® * . 
constellations, and especially to provide for the stars near the southern p^oiemaic 
pole. Many other constellations have been proposed at one time or 
another, but have since been rejected as useless or impertinent, though 
about a dozen have obtained partial acceptance and still hold a place upon 
some star-maps. 

Originally certain stars were reckoned as belonging to more Consteiia- 
than one constellation, but at present this is no long-er the case : ^^^.^ ^°"° ' 

' ^ o aries. 

the entire surface of the celestial sphere is divided up between 
recognized constellations. There is, however, no decisive defi- 
nition of their respective boundaries, and different authorities 
disagree at many points. Argelander is now generally accepted 
as the authority for the northern constellations and Gould for 
the southern. 

A thorough knowledge of these artificial star groups and of the names Knowledge 

and places of the stars that compose them is not at all essential, even to an ^^ constella- 

accomplished astronomer ; but it is a matter of great convenience and of , , ' ' 

. . . . . ■ able, but 

real interest to an intelligent person to be acquainted with the principal j^^^ essential 

constellations ^ and to be able to recognize at a glance the brighter to an 
stars, — from fifty to one hundred in number. This amount of knowledge astronomer. 
is easily obtained in a few evenings by studying the heavens in connec- 
tion with a good celestial globe or star-map, taking care, of course, to 
select evenings in different seasons of the year, so that the whole sky may 
be covered. 

532. Methods of designating Individual Stars. — (a) By Names. Designation 
About sixty of the brighter stars have names in more or less of stars: by 

•^ ^ names. 

common use. 

1 In his Uranography, a booklet of about fifty pages, published by Ginn & 
Company, the author has given a brief description of the various constellations 
and directions for tracing them. The star-maps which accompany it are quite 
sufficient for this purpose, though not on a scale large enough to answer for 
detailed study. For reference purposes, Professor Upton's Star Atlas (issued 
by the same publishers) is recommended, or the still more elaborate (and expen- 
sive) ones of Argelander, Heis, or Klein. 



By place in 

By constel- 
lation and 

A majority of these names are of Greek or Latin origin {e.g.^ 
Capella, Sirius, Arcturus, Procyon, Regulus, etc.); others have 
Arabic names (Aldebaran, Vega, Rigel, Altair, etc.). For the 
smaller stars the names ^ are almost entirely Arabic. 

(b) By the Star 8 Place in the Constellation. This was the 
usual method employed by Ptolemy and Tycho Brahe. 

Spica^ for instance, is the star in the spike of wheat which 
Virgo carries ; Cynosure is Greek for " the tail of the dog " (in 
ancient times the constellation which we now call Ursa Minor 
was a dog); Capella is the goat which Auriga, the charioteer, 
carries in his arms. Hipparchus, Ptolemy, in fact all the 
older astronomers, including Tycho Brahe, used this method to 
indicate particular stars, speaking, for instance, of " the star 
in the head of Hercules," or in the "right knee of Bootes" 

(o) By Constellation and Letter. In 1603 Bayer, in publish- 
ing his star-map, adopted an excellent plan, ever since followed, 
of designating the stars in a constellation by the letters of 
the Greek alphabet. The letters generally (not always) were 
applied in the order of brightness, a being the brightest star of 
the constellation and /3 the next brightest; but they are some- 
times (as in the case of " The Dipper ") assigned to the stars in 
their order of position rather than in that of brightness. 

When the naked-eye stars of a constellation are so numerous 
as to exhaust the letters of the Greek alphabet the Roman 
letters are next used, and then, if necessary, we employ num- 
bers which Flamsteed assigned a century later. 

At present every naked-eye star can be referred to and iden- 
tified by its letter or Flamsteed number in the constellation to 
which it belongs. 

1 Allen's Star-Names and their Meanings (G. E. Stechert Company, New- 
York) is the best work on the subject ; full of curious and interesting informa- 
tion relating to the names themselves, and to the vaxious legends connected with 
them and with the constellations. 


{d) By Catalogue Number. The preceding methods all fail Bynmuber 
in the case of telescopic stars. To such we refer as number "^^^^^^■ 

^ catalogue. 

so-and-so of some one's catalogue; thus, "LI., 21185 " is read 
"Lalande, 21185," and means the star so numbered in Lalande's 
catalogue. At present about eight hundred thousand different 
stars are contained in our numerous catalogues, so that (except 
in the Milky Way) ever}^ star visible in a 3-inch telescope can 
be found and identified in one or more of them. 

Syno7iyms. Of course all the bright stars which have names Synonyms. 
have letters also and are sure to be found in every catalogue 
which covers their part of the heavens. A star notable for any 
reason has, therefore, usually many " aliases," and sometimes 
care is necessary to avoid mistakes on this account. 

533. Star-Catalogues These are lists of stars, arranged in Ancient and 

some regular order, giving their positions (z.e., their right ascen- ^^^^'^^^'^^ 
sions and declinations, or longitudes and latitudes), and usually 
also indicating their so-called magnitudes or brightness. 

The first of these star-catalogues was made about 125 B.C. by 
Hipparchus of Bithynia (the first of the world's great astrono- 
mers), giving the longitude and latitude of 1080 stars. This 
catalogue was republished by Ptolemy 250 years later, the 
longitudes being corrected for precession, though not quite 

The next of the old catalogues of any value was that of 
Ulugh Beigh, made at Samarcand about a.d. 1450. It was 
followed in 1580 by the catalogue of Tycho Brahe, containing 
1005 stars, the last constructed before the invention of the 

The modern catalogues are numerous, — already counted by 
the hundred. Some give the places of a great number of stars 
rather roughly, merely as a means of identifying them when 
used for cometary observations or other similar purposes. To The 


heavens, which contains over 324000 stars, — the largest number ungs. 

this class belonofs Argelander's Durchmusterunq of the modern 

^ ^ muster 





The Gesell- 
schaft cata- 

mental star 
places de- 
termined by 

places by 

in any one catalogue thus far published. This has since been 
supplemented by Schoenf eld's Southern Durchmusterung^ on 
a similar plan. 

Then there are the " Fundamental Catalogues," like the Pul- 
kowa and Greenwich catalogues, which give the places of a few 
hundred stars only, but as accurately as possible, in order to 
furnish reference points in the sky. 

The so-called " Zones " of Bessel, Argelander, Gould, and 
many others are catalogues covering limited portions of the 
beavens, containing stars arranged in zones about a degree 
wide in declination and running through some hours in right 

An immense catalogue is now in process of publication under 
the auspices of the German Astronomische Gesellschaft, and 
will contain accurate places of all stars above the ninth mag- 
nitude north of 15° south declination. The observations, 
by numerous cooperating observatories, have occupied twenty 
years, but are at last finished, and three quarters of the dif- 
ferent parts of the catalogue are already published. The Cor- 
dova catalogue and Cordova " Zones," together with the cata- 
logues and Photographic Bur chmuste rung of the Cape of Good 
Hope Observatory, cover the rest of the southern heavens. 

534. The Determination of Star Places for Catalogues The 

observations from which a star-catalogue is constructed have 
until lately been usually made with the meridian-circle. For the 
fundamental catalogues comparatively few stars are observed, 
but all with the utmost care and on every possible opportunity, 
during several years, with every precaution to eliminate all 
instrumental and observational errors. 

In the more extensive catalogues most of the stars have been 
observed only two or three times, and everything is made to 
depend upon the accuracy of the places of the " fundamental 
stars," which are assumed as correct. The instrument in this 
case is used only "differentially" to measure the comparatively 


small difference between the right ascension and declination of 
the fundamental stars and those of the stars to be catalogued. 

At present, by means of photography, the catalogues are Photog- 
beinsr extended to stars much fainter than those observable by ^'^^^]^ "*^^ 

o ^ ^ '^ used. 

meridian-circles. On the photographic plates the positions of 
the smaller stars are determined by reference to larger stars 
which appear upon the same plate, and the catalogues now in 
process of construction from the photographic campaign will 
contain between one and two million stars down to the eleventh 

535. Mean and Apparent Places of the Stars. — The modern Reduction 
star-catalogfue contains the mean rig-ht ascension and declination °, ^^^^ 

° _ *=* ^ place to ap- 

of its stars at the beginning of some designated year, i.e.^ the parent place 
place the star would occupy on that date if there were no ^^^ ^^^^ 

. , , versa, 

equation of the equinoxes, nutation, aberration, or j^'^oper motion. 

To get the actual (apparent) right ascension and declination of 
a star for some given date (which is what we always want in 
practice), the catalogue place must be "reduced" to that date, 
i.e., it must be corrected for precession, aberration, etc. The 
operation with modern tables and formulse is not a very tedious 
one, involving perhaps five minutes work, but without it the 
catalogue places are useless for accurate purposes. Viee versa, 
the observations of a fixed star with the meridian-circle do not 
give its mean right ascension and declination ready to go into 
the catalogue, but the observations, before they can be tabu- 
lated, must be reduced backwards from the apparent place 
observed to the mean place for some chosen '' epoch." 

536. Star Charts and Stellar Photography. — For certain pur- star charts, 
poses accurate star charts are even more useful than catalogues. 

The old-fashioned way of making such charts was by plotting 
the results of zone observations, but at present it is being done, 
by means of photography, vastly better and more rapidly. A The photo- 
cooperative campaiscn beg^an in 1889, the obiect of which is to s^'^P^^^^ 

•^ J. o o J campaign. 

secure a photographic chart of all the stars down to the 



No limit yet 
found to 
faintness of 
stars tliat 
can be pho- 

The Paris 
graphic tele- 

fourteenth magnitude. The work is now (1902) fully three 
fourths done. Eighteen different observatories have participated 
in the worii. From these chart plates extensive catalogues are 
also being made, as already mentioned. 

One of the most remarkable tilings about the photographic 
method is that with a good instrument there appears to 

be no limit to the faintness 
of the stars that can be photo- 

graphed; by increasing the 
time of exposure, smaller and 
smaller stars are continually 
reached. With the ordinar}^ 
plates, and exposure times not 
exceeding twenty minutes, it 
is now possible to get distinct 
impressions of stars that the 
eye cannot possibly see with 
tlie telescope employed. 

Fig. 173 is a representation 
of the Paris instrument of the 
Henry Brothers, which was 
the first employed in such 
work and was adopted as the 
typical instrument for the 

Fig. 173. -Photographic Telescope of the charting operation. It has an 
Pans Observatory ° ^ 

aperture of about 14 inches 
and a length of about 11 feet, the object-glass being speciall}^ 
corrected for the photographic rays. A 9-inch visual telescope 
is inclosed in the same tube, so that the observer can watch 
the direction of the instrument during the whole operation. 

The instruments used at the other observatories differ in 
mechanical arrangements, but all have lenses of the same 
aperture and focal length, the scale of all the photographs being 
1' to a millimeter, — the same as that of Argelander's charts. 



As already mentioned, these charts furnish the material for a 
very extensive catalogue. 

Several other very large photographic telescopes have already been Other large 
constructed. The Bruce telescope, presented to the Harvard College photo- 
Observatory by the late Miss Bruce of New York, has for its objective a S^'^P^"*^ ^^^6- 
four-lens photographic doublet 2 feet in diameter, but with a focal length 
of only 1 1 feet, — the same as those mentioned above, — so that its nega- 
tives are on the same scale. While the ordinary photographic lens will 
cover an area of only about two degrees square, this covers from five to six 
degrees square, and with a very much diminished time of exposure. It has 
been sent to the Harvard subsidiary observatory at Arequipa, Peru, where 
it is employed in the photography and spectroscopy of the southern heavens. 

The new telescopes at Greenwich and the Cape of Good Hope have the 
same aperture, but are much longer. Both have visual finders 18 inches 
in diameter. 

The enormous instrument at Meudon (near Paris) has also two tele- 
scopes combined, — a "v^isual telescoj^e of 32 inches aperture and a photo- 
graphic of 25 inches, each 55 feet focal length. 

Still more recently the Potsdam Astrophysical Observatory has mounted 
an immense instrument, shown in the frontispiece, the photographic object- 
glass of which has a diameter of 31^ inches, with a focal length of 43 feet, 
and the visual object-glass a diameter of 20 inches. This long-focus instru- 
ment will, however, be used mainly for other purposes than charting. 


537. In contradistinction from the planets, or "wanderers," 
the stars are called "fixed," because they keep their relative The so- 
positions and config-urations sensibly unchangfed for centuries. ^^^^^^ ^^^^ 
Delicate observations, however, separated by surhcient intervals moving, 
of time, show that the fixity is not absolute. Nearly two 
hundred years ago (in 1718) it was discovered by Halley that 
Arcturus and Sirius had changed their places since the days of 
Ptolemy, having moved southward, the first by a full degree 
and the other about half as much. Indeed, even to the naked 
eye, these two stars no longer fit certain alignments described 
by Ptolemy. 



motions due 
to earth's 
only ap- 

by com- 
parison of 
old with 
recent star- 

Modern observations show clearly that the stars are really all 
in motion, "drifting" upon the celestial sphere. Not only so, 
but the spectroscope now makes it possible to measure their rate 
of motion towards or from the earth, and it appears on the 
whole that their velocities are of the same order as those of 
the planets : they are flying through space incomparably more 
swiftly than cannon-shot, and it is only because of their incon- 
ceivable distance from us that they seem to go so slowly. 

538. Common Motions. — If we compare a star's position (i.e., 
its right ascension and declination) as determined to-day by a 
meridian-circle with that observed one hundred years ago, it will 
always be found to have altered considerably. The change, how- 
ever, is mainly due, not to any real change in the position of the 
star, but to precession, nutation, and aberration, already dis- 
cussed (Sees. 165-171). 

These depend upon variation in the direction of the earth's 
axis and upon tlie swiftness of her orbital motion and are not 
real changes of the star's direction from the earth. They are 
only apijarent displacements and are called " common " motions 
because they are shared alike by all stars in the same region of 
the sky. They do not in the least affect their apparent con- 
figurations and angular distances from each other. 

539. Proper Motions. — But after allowing for all these 
common motions of the stars, it generally appears that in the 
course of a century the stars have really changfed their places 
with reference to each other^ each having a motus pecuUaris, 
or "proper motion" of its own, the word "proper" being here 
the antithesis of " common." Of two stars side by side in 
the same telescopic field of view the proper motions may be 
very different in amount, or even directly opposite, while the 
common motions, due to precession, etc., are, of course, sensibly 

About 175 stars are at present known to have a proper motion 
exceeding 1'' annually, but the number is being constantly 


increased by additions from among the fainter stars. Even 
the largest of these proper motions (always expressed in seconds 
of arc) is very small. 

The maximum at present known (discovered in 1898) is that of a little Maximum 
star of the eighth magnitude, known as " G.C.Z., V, No. 243 " (i.e., Gould's kuowu S'\7 
Cordova Zones, Fifth Hour, No. 243), which drifts 8".7 yearly. The next y^^^^^' 
in magnitude, and for a long time at the head of the list, is that of the 
seventh-magnitude star, 1830 Groombridge, the so-called " runaway star," 
which has an annual drift of T\ Neither of these stars is visible to the 
naked eye. It will take two hundred years for the first of them to drift a 
distance equal to the moon's apparent diameter. 

As might be expected, the proper motions of the bright stars Average 
average higher than those of the faint ones, since, on the whole, "^°^^J^^, 
the bright stars are nearer ; but the faint stars are so much the nearer 
more numerous that among them many drift faster than any of ^^^^^' 
the fewer bright ones. 

The average proper motion of the first-magnitude stars is 
about ^" annually, and that of the sixth-magnitude stars (the 
smallest visible to the naked eye) is about ^L of a second. 

These motions are always sensibly rectilinear. 

Table IV of the Appendix, in connection with other matters, gives the 
proper motions of about forty of the nearer stars which also, as a rule, are 
the stars having the larger proper motions. 

Hitherto the determination of proper motions has rested Advantage 
almost entirely upon the comparison of remotely dated star- ofphotog- 

... . raphy in 

catalogues, but it is likely that hereafter much more rapid determining 
progress will be made by the comparison of photographic charts, P^'oper 
in which consideration of the common motions is unnecessary, 
as these affect alike all the stars on each negative. 

540. Real Motions of Stars. — The proper motion of a star Real motion 
gives us very little knowledge as to the star's real motion in °^ ^ ^^^^' 
miles unless we know the star's distance, nor even then unless 
we also know its rate of motion towards or from us. The 



of proper 
motion to 
miles re- 
of distance. 

for cross 

To the Earth 

Fig. 174. 

— Components of a Star's 
Proper Motion 

proper motion derived from the comparison of the catalogues of 
different dates is only the angular value of that part of the whole 
motion which \^ perpendicular to the line of vision^ the "cross" 
or " thwartwise " motion, as it may be called. A star moving 
directly towards or from the earth has no proper motion, 

I.e., no change of apparent place 
to be detected by comparing 
observations of its position. 

Fig. 174 illustrates the mat- 
ter. If a star really moves in a 
year from A to B^ it will seem 
to an observer at the earth to 
have traversed the line Ah^ and the proper motion (in seconds 


of arc) will be 206265 x -n Since Ah cannot possibly 


be greater than AB^ we are able in some cases to fix a minor 

limit to the star's velocity. 

According to the determination of Briinnow, accepted until lately, the 

distance of 1830 Groombridge is a little over two million astronomical 

units; and therefore, since Ah subtends an angle of 1'' at the earth, its 

7x2 000000 

length must be at least astronomical units, which, reduced 

^ 206265 

to miles and divided by the number of seconds in a year, corresponds to a 

velocity exceeding 200 miles a second. 

More recent observations by Kapteyn make the distance of this star con- 
siderably less — about 1400000 astronomical units — -and proportionally 
reduce the cross motion. Ah, to about 140 miles a second. 

For the star of greatest proper motion, G.C.Z., V, 243, the cross motion 
comes out about 80 miles per second, so that the " runaway star " still 
holds the record for real swiftness. 

The formula for this "cross " or "thwartwise " motion (^Ah in Fig. 174) is 


© (miles per second) = 2.944 - = 0.903 y x ix, 


where /x is the annual proper motion of the star, p its parallax (both in 

seconds of arc), and y its distance in "light-years." (See Sees. 546 

and 547.) 


In many cases a number of stars in the same region of the 
sky have proper motions practically identical, making it almost 
certain that they are in some sense neighbors and really con- 
nected, — very likely by community of origin. In fact, it seems Gregarious 
the rule rather than the exception that stars which are arjpar- tendency of 


ently near each other and about alike in brightness are really 
comrades. They show, as Miss Gierke expresses it, a distinctly 
"gregarious" tendency. In certain cases, however, there are 
groups of stars in which some conspicuous members have dif- 
ferent proper motions from the others, and these discordant 
motions will in time destroy the configuration. The " Dipper " 
of iJrsa Major is a case in point. The two extreme stars, a Proper 
and 77, are, according to Flammarion, moving in a nearly oppo- ^^^^tious m 

' \ , ^ ^ J J/i Ursa Major. 

site direction from the others, so that about one hundred 
thousand years ago the " Dipper " was no dipper at all, and will 
not be one a hundred thousand years hence. The other stars of 
the group maintain their configuration. 

541. Motion in the Line of Sight, or << Radial Velocity.** ^^ Spectro- 
Observations of the proper motions of stars furnish no infor- scopic deter- 

, . , - ^ . mination of 

mation as to the rate at which the stars are receding or radial 
approaching ; but if a star is bright enough to give an observ- velocity. 
able spectrum, its radial velocity can be determined by means 
of the spectroscope and the application of the Doppler-Fizeau Application 
principle (Sec. 254). If the star is receding, the lines of its °J ^^^^ 
spectrum will be shifted towards the red, and towards the blue Fizeau 
if it is coming nearer. The shift is ascertained by arranging the pi'i^cipie. 
telespectroscope (Sec. 244) so that by a comparison prism the 
observer shall have, close together or superposed, the spectrum of 
the star he is dealing with and of some substance (hydrogen, 
sodium, iron, or titanium) whose lines are present as dark lines in 

1 We shall follow the French usage in employing the term " radial velocity " 
{vitesse radiale) to denote the rate at which a body is changing its distance from 
the observer. The equivalent expression, "motion in line of sight," is rather 



First success 
by Huggins 
in 18G7. 


work on 

of photog- 


the star spectrum ; he can then appreciate and measure any dis- 
placement of the stellar lines, as illustrated by Fig. 98, Sec. 254. 
Sir William Huggins, in 1867, was the first to apply this 
method, and obtained some very interesting results (especially 
the determination of the radial motion of Sirius), quite sufficient 
to establish the feasibility of his method. From the insufficient 
power of his instruments, however, they can now be regarded 
only as approximations. 

The work was followed up for several years at Greenwich 
and some other places, but so long as visual observations were 

depended upon 
the results were 
not very satisfac- 
tory. Visual olj- 
servations of this 
kind are ex- 
tremely difficult; 
the star spectra 
are very faint, the 
displacements of 
the lines very 

Fig. 175. — Spectrum of a Aurigae compared with Hydrogen minute, and the 

^^sei lines themselves 

often broad and hazy and ill adapted for accurate measurement. 

In the case of the nebulse, however, which give spectra con- 
taining sharp, bright lines. Professor Keeler of the Lick Observa- 
tory has made visual observations which fairly compete with 
photographic work. 

542. Spectrographic Determination of Radial Velocity. — The 
unsatisfactory results of visual observations led Vogel in 1888- 
89 to apply photography, and with immediate success. In this 
case the difficulties arising from the faintness of the star spectra 
can be largely overcome by prolonged exposure, and all necessary 
measurements can be made at leisure under the microscope. 


Fig. 175 (borrowed by permission from Frost's translation of Scheiner's 
Astronomical Spectroscopy') shows very perfectly the actual appearance of 
part of the negative of the spectrum of a Aurigse (Capella) and the corre- 
sponding part of the solar spectrum as seen under the microscope with 
which the measurements are made. The solar spectrum is, of course, on 
a separate plate, but this plate and the star negative are clamped together 
so as to make the lines correspond and facilitate the identification of lines 
in the star spectrum. (ISTote in passing the perfect correspondence between 
the spectrum of this star and that of the sun.) The sharp black line 
which crosses the narrow star spectrum is the "Hydrogen y" bright line 
in the spectrum of a Geissler tube placed in the cone of rays about 2 feet 
above the slit-plate and illuminated by electricity for a few seconds at 
different times during the long exposure (an hour or so) which is required 
for the star spectrum. 

One sees easily that in this case the star line is shifted slightly to the 
right, but it appears to be so poorly defined that accurate measurement 
would be difficult. For the methods by which this difficulty is overcome, and 
for the corrections required on account of the motion of the eartli and other 
causes, the reader is referred to the book from which the figure is taken. 

Fig. 176 is from a photograph of the new Potsdam spectrograph, 
attached to the great telescope shown in the frontispiece. It is a much 
more powerful and perfect instrument than that used by Yogel in the 
work above mentioned. 

In 1889 the probable error of a determination of radial velocity was 
about a mile a second ; at present (1902) it does not exceed one fourth of 
a mile for the best observers and instruments under favorable conditions. 

Table V of the Appendix presents the results of Vogel for 
the fifty-one stars that he had been able to deal with up to 
1892, with the addition of one or two from other observers. 
His telescope had an aperture of only 11 inches, wliich limited 
him to the brighter stars. It has now been replaced by the 
much larger instrument, figured in the frontispiece. 

The maximum velocity indicated by his observations of radial 
1888-89 is that of a Tauri, 30.1 miles a second, receding. ^^^^^^^^ 

. . . * so far 

The next in order is that of <y Leonis, 24.1 miles, approaching, measured 
Belopolsky, at Pulkowa, has since found for ^ Hercules the ^^outfio 

miles n 

higher velocity of 44 miles, approaching ; and Campbell, at the second. 




Lick Observatory, finds for ^ Cassiopeise the still higher velocity 
of 61 miles, also approaching, and for 8 Leporis and 6 Canis 
Majoris a nearly equal receding velocity. 

Since 1890 the same line of work has been taken up success- 
fully by many observers, especially by Belopolsky at Pulkowa, 

Fig. 176. — The New Potsdam Spectrograph 

and in this country by Keeler and Campbell at the Lick 
Observatory, and by Frost at the Yerkes.^ Fig. 177 is enlarged 
from one of Frost's photographs of the spectrum of Polaris 
(a positive in this case) compared with that of the metal titanium^ 

1 See Fig. 180, on page 505, for the Yerkes spectrograph. 



which has been found specially advantageous for such compari- 
sons. In this case, the radial velocity of the star is so small 
(about 8 miles a second) that the shift of the lines in the 
figure is hardly perceptible to the eye. 

Observations by Humphreys and Mohler of Baltimore in Causes 
1895 (already mentioned in Sec. 256) show that under heavy ^^^"!!^,^ 

\ 'J ' ^ -J modify 

pressure the spectrum lines of many elements shift slightly results for 
towards the red, very much as if the luminous object were ^^^^^^ 



. ■ • 



*- ■% « ' #i # «^ - ^i»fe-^^#^iii^#K^ 

' 1 

^'1 ' 1 









Fig, 177. — Spectrum of Polairis compared with Titanium 


receding. The shift under a given pressure is, however, dif- 
ferent for different substances and for different lines of the 
same substance. It is always minute, never, even under a 
pressure of ten or twelve atmospheres, exceeding the displace- 
ment that would be due to a receding velocity of 1 or 2 miles 
a second, but it is quite sufficient to require to be examined 
and taken into account in all applications of Doppler's principle. 

543. The Sun*s Way, or Motion of the Solar System. — The The sun's 
sun, like other "stars," is traveling through space, taking with "motion m 
it the earth and the planets. 



tion of its 
direction by 
means of 
motions of 

tion by 

Apex of the 
sun's way : 
mate posi- 

Sir William Herschel was the first to investigate and deter- 
mine the direction of this motion, more than a century ago. 
The principle involved is this : the apparent motion of a star 
relative to the sun is made up of its own real motion combined 
with the sun's motion reversed. If the stars, therefore, were 
absolutely at rest, they would apparently all drift bodily in a 
direction opposite to the sun's real motion. If, as is the fact, 
they themselves are in motion, and if their motions are indis- 
criminately in all possible directions (an assumption probable as 
an approximation to the truth, but which can hardly be proved 
as yet), there will be, on the whole^ a similar drift. Those in 
that quarter of the sky towards which we are approaching will, 
on the whole, open out from each other, and those in the rear 
will close up behind us, while in the region of the sky between, 
they will, on the whole, drift backwards, — just as one walking 
in a park filled with people moving indiscriminately in different 
directions would, on the whole, find that those in front of him 
appear to grow larger,^ and the spaces between them to open 
out, while at the sides they would drift backwards, and in the 
rear close up. 

Again, from the radial motions of the stars spectroscopically 
measured a result can be obtained. In the portion of the 
heavens towards which the sun is moving the stars will, on the 
whole, seem to approach, and in the opposite quarter to recede. 

The individual motions, proper and radial, lie in all direc- 
tions ; but when we deal with them by the thousand the indi- 
vidual is lost in the general, and the prevailing drift appears. 

About twenty different determinations of the point in the 
sky towards which the sun's motion is directed have been thus 
far made by various a^stronomers. There is a reasonable accord- 
ance of results, and they all show that the sun, with its 

1 Theoretically, of course, the stars towards which we are moving must 
appear to grow brighter as well as to drift apart, but this change of brightness, 
though real, is entirely imperceptible within a human lifetime. 


attendant planets, is moving towards a jjoint on the borders of the 
constellation of ITercides, having^ according to New comb ^ a right 
ascension of about 277°. 5^ and a declination of about 35°. This 
point is called the Apex of the suit's way. 

There is, however, a curious systematic difference between uncertainty 
the results obtained by comparing the proper motions of stars ^^ to exact 
that drift very slightly with those that drift more rapidly. 
Dividing them into four groups, some 550 of the slower stars 
give for the " apex " a declination of about 42°, those of the 
next grade of about 40°, those still nearer about 35°, while 
those that are nearest us, or at least have the largest proper 
motion, push the point still nearer to the equator, to a declina- 
tion of about 30°. The right ascension deduced for the apex 
shows no such systematic discordance. 

This probably indicates that the motions of the stars are not 
absolutely indiscriminate, but that those that are near to us 
have some common drift of their own. 

As to the velocity of the sun's motion in space, the spectro- Velocity 
scopic results, which are on the whole more trustworthy since ^^^^^^^ ^■'■ 

■^ ^ ^ _ -^ _ miles a 

they involve no assumption as to the distance of the stars, indi- second, 
cate that it is about 11 iiiiles a second., which probably is very 
near the truth. 

544. The Imagined << Central Sun.'' — We mention this sub- No central 
ject simply to say that there is no satisfactory foundation for 
the belief in the existence of such a body. The idea that the 
motion of our sun and of the other stars is a revolution around 
some great central sun is a very fascinating one to certain minds, 
and one that has been frequently suggested. It was seriously 
advocated half a century ago by Maedler, who placed this center 
of the universe at Alcyone, the principal star in the Pleiades. 

It is certainly within bounds to deny that there is any con- 
clusive evidence of such a motion, and it is still less probable 
that the star Alcyone is its center, if the motion exists. (But 
see Sec. 609, last paragraph.) 




The stellar 
system a 

So far as we can judge at present, it is most likely that the stars are 
moving, not in regular closed orbits around any center whatever, but 
rather as bees do in a swarm, — each for itself, under the action of the 
predominant attraction of its nearest neighbors. The solar system is an 
absolute monarchy, with the sun supreme. The great stellar system appears 
to be a republic, without any such central and dominant authority. 

or annual 

twice a 

of star's 

star's dis- 
tance and 


545. Heliocentric or Annual Parallax. — This has already 
been defined (Sec. 78) as the difference between the direction 
of a star seen from the sun and from the earth, which difference, 
if the star is not at the pole of the ecliptic, varies through- 
out the year with an annual periodicity. In the case of a star 
the geocentric or "diurnal" parallax is absolutely insensible, 
— hopelessly beyond all present power of measurement. 

When, therefore, we speak of a stars parallax the heliocen- 
tric parallax is to be understood. Moreover, unless otherwise 
clearly indicated, the maximum value of the star's heliocentric 
parallax is always meant. Twice a year the earth is so situated 
that the sun and star are 90° apart in the sky, when the longi- 
tude (celestial) of the sun is 90° greater or less than that of the 
star. At that moment the radius vector of the earth's orbit is 
perpendicular to a line drawn from the earth to the star, and 
the star's annual parallax has its greatest possible value. 

The parallax of a star may therefore be defined, as the term 
is ordinarily used, to be the angle subtended hy the semi-major 
axis of the earths orbit when viewed perpendicularly from the 
star. In Fig. 178, R is the distance from the earth to the sun, 
D from the sun to the star, and the angle p is the star's paral- 
lax. If we can measure p^' (i.e., p in seconds of arc), the dis- 
tance of the star at once follow^s from the equation 

D = 





sm j9" p' 

R being the radius of the earth's orbit, 93 000000 miles. 


546. The Starts Parallactic Orbit. — We may look at the Parallactic 
matter differently. In accordance with the principle of relative ^^^^^ ®* * 

-^ . . star. 

motion (Sec. 354), every star really at rest (leaving aberration 
out of the account at present) must appear to us to move in 
a little "parallactic orbit" 186 000000 miles in diameter, the 
precise counterpart of the earth's orbit, and having its plane 
parallel to the ecliptic ; in this little orbit the star keeps always 
opposite to the earth. If the star is at the pole of the ecliptic, 
we see this parallactic orbit as a circle ; if in the ecliptic, edge- 
wise, as a short straight line ; in intermediate (celestial) lati- 
tudes, as an oval. The semi-major axis of this apparent paral- 
lactic oi'bit is, of course, the star's parallax. (In Fig. 178 the 

Fig. 178. — Heliocentric Parallax 

dotted oval is the parallactic orbit of the star s, as seen from 
the solar system.) If the star is drifting (proper motion), this 
motion will, of course, be combined with the parallactic move- 
ment, but the two can easily be separated by calculation. 

For a Centauri, which is our nearest neighbor among the Maximum 
stars so far as we know at present, the parallax is only 0''.75, p^^*^^!^^^* 

■^ , -^ *^ present 

and there are but seven other stars which are known to have a known, 
parallax as large as O^'.S. Indeed, the whole number of those o"-75for 
which are fairly determined to have a parallax of O'M and over 
is less than forty. 

547. Unit of Stellar Distance; the Light- Year. — The distances The light- 
of the stars are so enormous that the radius of the earth's ^^^^' 
orbit, the astronomical unit hitherto employed, is too small 
for a convenient measure. It is better, and now usual, to 


ma:n"ual of astronomy 

Formula for 
distance in 

No star as 
near as three 

Difficulty of 

success in 

take as the unit of stellar distance the so-called ''light-year," 
i.e., the distance which light travels in a year, — about sixty- 
three thousand ^ times the distance of the earth from the sun. 

A star with a parallax of one second is at a distance of 
206265 astronomical units. Dividing this by 63000, we find 

its distance in light-years, 7>,^ = ^-^ • 

A star with a parallax of half a second is at a distance of 

6.52 light-years, and a star with a parallax of one tenth of a 

second is at a distance of 32.6 light-years. 

So far as can be judged from the scanty data available, it 
appears that few, if any, stars are within a distance of three 
light-years from the solar system, — not one has thus far been 
discovered ; that the naked-eye stars are probably, for the most 
part, within two or three hundred light-years; and that many 
of the telescopic stars must be some thousands of years away. 

Table IV of the Appendix contains a list of the parallaxes 
thus far best determined. 

548. The Determination of Stellar Parallax. — It is obvious, 
therefore, that while simple enough in principle, the measure- 
ment of a star's parallax is practically one of the most delicate 
and difficult of all astronomical operations ; and there is no way 
at present of evading the difficulty or " flanking the position," 
so to speak (as in the case of the solar parallax), by measuring 
the parallax of some near object or utilizing our knowledge of 
the speed of light. 

Many attempts were made by early astronomers to measure 
the parallax of stars, but with no real success until Bessel, in 
1838, succeeded in determining the parallax of 61 Cygni, a 
little star of the sixth magnitude, which had for some time been 

1 This number is found by dividing the number of seconds in a sidereal year, 
31 558149, by 499, the number of seconds required by light to travel from the 
sun to the earth. The exact quotient is 63243. The light-year bears to the 
astronomical unit almost exactly the same ratio as the mile to the inch. 


interesting astronomers on account of its great proper motion 
of 5" a year. He made his observations with a heliometer and 
ascertained its parallax to be about half a second, but more 
recent determinations bring it down to 0''.40. 

Almost simultaneously Henderson announced the parallax of Henderson 
a Centauri as 0".9, according to meridian-circle observations ^"^^'^c®^^" 

^ tauri. 

at the Cape of Good Hope. The star has a large proper 
motion and is one of the brightest in the heavens, but is not 
visible in our latitudes. It still holds its place as our nearest 
neighbor, though later observations show that its parallax is 
only 0''.75. 

Two methods of procedure have thus far been used, known 
as the absolute method and the differential method. 

549. The Absolute Method. — This consists in making with a The abso- 
meridian-circle, or some equivalent instrument, numerous obser- ^^^® ^^^®* ^° * 
vations of the star's apparent right ascension and declination 
throughout the year, especially at the two seasons when the 
parallax has its largest value. These observations are then care- 
fully corrected for aberration, precession, and nutation, also for 
the star's proper motion, and for any known errors due to the 
action of the seasons on the instrument. If the observations 
and their corrections are perfectly accurate, they will give a 
set of slightly different positions for the star during the year, 
which, when plotted, will all fall on the circumference of a little 
oval, — the star's "parallactic orbit." One half the angular 
length of the orbit will be the star's parallax. 

But the corrections to be applied to the observations are Embar- 
enormous compared with the parallax itself. While the paral- ^'^^^®^ ^y 

, ^ large correc- 

lax is only a fraction of a second, the aberration corrections run tions. 
up to 41''. The instrumental correction is especiall}^ trouble- 
some, because it runs its course yearly, just as the parallax does, 
and any outstanding error confounds itself with the parallax in 
a manner almost inextricable. Hence, comparatively little suc- 
cess has attended operations of this sort, though it was by such 



The difeer- 



observations that the parallax of a Centauri was first detected, 
as already stated. 

550. The Differential Method. — This consists in determining 
the position of the suspected star at different times during the 
year, not absolutely, but with reference to the smaller stars appar- 
ently near it, though presumably at a great distance beyond. This 
almost entirely obviates the difficulty due to aberration, preces- 
sion, and nutation, since these affect all the stars concerned in 
the operation nearly alike ; the observations therefore need cor- 
rection only for the difference between the aberration, etc., of 

Fig. 179. — Differential Measurement of Parallax 

It avoids 
large cor- 

the investigated star and that of each of its neighbors, and this 
small differential correction can be easily computed with very 
great accuracy. To a considerable extent also the method 
evades the effect of refraction and that of temperature disturb- 
ances upon the instrument, since any displacement of the instru- 
ment does not perceptibly affect the relative position of the stars 
seen through it. Per contra^ the method measures, not the whole 
parallax of the star investigated, but only the difference betwee^i 
its parallax ayid that of the stars with which it is compared. 

Suppose, for instance (Fig. 179), that in the same telescopic 
field of view we have the star s, which is near us, the stars 


X and I/, which are so remote that they have no sensible parallax 
at all, and the star 2, which is more remote than s but has a 
sensible parallax of its own. s and z will describe their paral- 
lactic orbits every year, just alike in form, and parallel, but the 
orbit of s will be much larger than ^'s. If now, during the 
year, we continually measure the distance and the direction from 
X OY y to the apparent places of s and z in their parallactic orbits, Parallax 
the results will sfive us the true dimensions of their two orbits. ^®^^^^®J^ 

° _ IS only the 

If, however, we had taken z as the reference point from which difference of 
to measure the parallactic motion of s, we should have found Parallax 

. . between 

less than the true value, as is obvious from the figure. Con- ^tars. 
sidering only the measures made at the moments when s and z 
are at the extremities of their parallactic orbits, the lines really 
measured from the star 2 to s will be ea and/6. If we assume 
that the parallax of z is insensible, i.e.^ that its parallactic orbit 
IS a mere point, these measured lines must be used in computa- 
tion as if they were reckoned from the point z and were zc and 
zd ; the major axis of the parallactic orbit of s would then come 
out as cd instead of ah^ and the computed parallax cs will be less 
than the true parallax as by the amount ac^ which equals ez^ the 
parallax of z. 

It follows that if the measurements are absolutely accurate. Parallax too 
the parallax deduced by this method can never be too large, ^™^i^- ^^s- 
but may be too small, — the distance of the star will be more great. 
or less exaggerated. If, however, the small reference stars are 
so remote that their parallax and proper motions are really 
insensible (z.e., less than 0'^01), the changes in the relative posi- 
tion of the star under investigation, after the correction for 
its proper motion has been applied, will be due simply to its 

551. Instrumental Work. — If the comparison stars are very instrument: 
near the one under investisfation, the necessary measure- fiianmcrom- 

° ^ "^ _ eter or 

ments may be made with the filar micrometer; but if the dis- heiiometer. 
tance exceeds a few minutes, we must resort to the heiiometer 



rapliy also 

parallax : 
stars nearer 
than the 

Large proper 
motion the 
best indica- 
tion of prob- 
able near- 

(Sec. 72), with which Bessel first succeeded, or we may employ 
photography, which the late Professor Pritchard at Oxford has 
done with some success. This has the advantage over the 
heliometer that the actual observations (in taking the photo- 
graphs) can be made much more rapidly than the heliometer 
measures, and the subsequent measurement of the photograph 
can be made under the microscope at leisure ; moreover, the 
suspected star can be compared with a considerable number of 
others, while the heliometer is usually limited to two or three, 
on account of the long time required to make each complete 
measurement. On the other hand, when the suspected star is 
much brighter than the reference stars its photograpliic image 
is so large and hazy as to render the measures of its position 
difficult and uncertain. 

On the whole, the differential method^ notwithstanding the 
fundamental objection which has been mentioned, is at present 
much more trustworthy than the absolute. 

Negative Parallax. Now and then it happens that the obser- 
vations appear to show a small negative parallax for the star. 
This may indicate simply insufficient accuracy of observation or 
computation, or, if the observations have been made by the 
differential method, it may mean that the investigated star is 
really beyond the comparison stars and therefore has a larger 
parallax than they, so that the difference between its parallax 
and that of the comparison stars comes out negative. Of course 
a real " negative parallax " is impossible. It would mean, as 
some one has said, that " the star is somewhere on the other 
side of nowhere." 

552. Selection of Stars to be examined for Parallax. — It is 
obviously necessary to choose for observations of this sort stars 
that are presumably near. The most important indication of 
proximity is a large proper motion. It is easy to see that if the 
observer were brought nearer to a star, the rapidity of its appar- 
ent drift across the sky would be increased ; and accordingly it 


is practically certain that the stars of large proper motion average 
nearer than those for which the motion is smaller, though the 
indication is not to be depended on in any individual case : 
if a star happened to be moving directly towards or from us, 
its proper motion would be zero, however near it might be. 
Brightness is, of course, also confirmatory, but nearly all the 
bright stars have been already attacked. Tlieir number is not 
very great, and the majority of them turn out to be much farther 
away than 61 Cygni. Among the millions of faint stars it is 
quite likely that some few individuals at least will be found 
nearer than even a Centauri. 

553. Possible Spectroscopic Method of the Future. — It will Parallax 
appear later that in the case of certain binary stars (Sec. 587), determmed 

^^ ^ .by spectro- 

which have the plane of the orbit nearly directed to the sun, scopic obser- 
the spectroscope will enable us to determine the actual speed ^^^lo^s of 

• 11-11 • 1- 1- 11 T • f l>iiiary stars. 

With which they move in this orbit and the true dimensions of 
the orbit in miles. Combining this with the apparent dimeTV- 
sions of their orbits in seco7ids of arc\ it will be possible to com- 
pute their distance far more accurately than by any direct 
measure of their parallax. But it will be many years yet before 
the necessary measures can become available in more than one 
or two instances, since the orbital periods of the -binaries which 
have an apparent orbit of measurable size are mostly long, 
ranging from twenty years to several hundred, and the orbits 
that lie in a favorable position are not numerous. 

554. General Conclusions. — It is obvious that the data so far Data too 
obtained are too scanty to warrant many general conclusions, scanty for 

r^ broad gener- 

We have not a sufficient number of well-defined parallaxes to aiizations. 

furnish a safe basis for averages, — say forty or fifty only, — and 

the smaller ones among them are subject to a probable error of 

at least twenty-five per cent. Consequently all calculations as 

to mass, brightness, etc., of individual stars are likely to differ 

widely from the truth. It is, however, already clear that, taken 

in classes, the stars of large proper motion average the nearest, 



Caution in 



results to 



the distance being roughly inversely proportional to the apparent 
rapidity of their drift ; and, further, that on this assumption the 
majority of the stars must be at a distance greater than fifty 
light-years, and the remoter stars in all likelihood many times 
more distant still. 

Something similar appears to be true in the relation between 
the brightness of the stars and their distance. 

Kapteyn, from stellar parallaxes already determined, also 
finds evidence of a roughly uniform distribution of stars in 
space, leaving the Milky Way out of the question. 

Another interesting fact appears, viz., that the stars which 
show a spectrum like the sun's are on the average much nearer 
than the so-called " Sirian " stars. 

But the student must be warned again that he cannot safely 
apply to individuals the results of such general averages. Any 
small star is not at all unlikely to be really brighter and larger 
than a bright star near it, just as in the case of two persons, 
one a youth and the other a man of mature age, it would be 
unsafe to predict from the mere difference of age which would 
have the longer life. 


1. Assuming the parallax of 61 Cygni as O'^AO, and that it is approach- 
ing the sun at the rate of 34.5 miles a second, how many years will it 
be before its brightness is increased 10 per cent by the diminution of its 
distance? Arts. 2050 years. 

2. Assuming the distance of 61 Cygni as 8.15 light-years, and that its 
radial and " thwartwise " velocities are 34,5 and 38 miles a second, respec- 
tively, find how near the star will come to the sun if it keeps up this motion 
uniformly, and how long it will take to reach this point of nearest approach. 

( Nearest approach, 6.03 light-years. 
( Time, 19900 years. 

3. Make the same calculation for a Aquilae, assuming its parallax as 
0".20, its proper motion as 0'^65 annually, and the rate at which it is 
approaching as 24 miles a second. 



4. What would be the time required to make the journey to Sirius 
(parallax 0".38) at the rate of 60 miles a second, and the fare at 1 cent a 
mile ? 

5. Deduce the formulae of Sec. 540, viz. 


© (miles per second) = 2.944 - = 0.903 y x ix, 


using the data of Sees. 546-547. 

Fig. 180. — Bruce Spectrograph of the Yerkes Observatory (1902) 



The Light of the Stars — Magnitudes and Briglitness — Color and Heat — Spectra- 

Variable Stars 

The six 
of naked- 
eye stars. 


555. Star Magnitudes — The term "magnitude," as applied 
to a star, refers simply to its brightness and has nothing to do 
with its apparent angular diameter. Hipparchus and Ptolemy 
arbitrarily graded the visible stars into six magnitudes, the stars 
of the sixth being the faintest visible to the eye, while the first 
magnitude comprises about twenty of the brightest. There is 
no known reason why six classes should have been constituted 
rather than eight or ten, unless perhaps the physiological one 
that ordinary eyes do not easily recognize differences of bright- 
ness sufficiently small to warrant a more refined subdivision. 

After the invention of the telescope the same system was 
extended to the smaller stars, but without any general agree- 
ment, so that the magnitudes assigned by different observers 
to telescopic stars in the early part of the century differ enor- 
mously. Sir William Herschel, especially, used very high num- 
bers, his twentieth magnitude being about the same as the 
fourteenth on the scale now generally used, which nearly corre- 
sponds with that which was adopted by Argelander in his great 
Durclimusterung (Sec. 533). 

Of course the stars classed together under one magnitude are 
not exactly alike in brightness, but shade from the bright to the 
fainter, so that exactness requires the use of fractional magni- 
tudes. It is now usual to employ decimals giving the bright- 
ness of a star to the nearest tenth of a magnitude. Thus, a 
star of 4.3 magnitude is a shade brighter than 4.4, and so on. 


thp: ltgpit of the stars 507 

A peculiar notation was employed by Ptolemy and used by Argelaiider Argelander's 
in his Uranometria ^ Nova. It recognizes thirds of a magnitude as the peculiar 
smallest subdivision. Thus, 2, 2,3, 8,2, and 3 express the gradations notation, 
between second and third magnitude, 2,3 being applied to a star whose 
brightness is a little inferior to the second, and 3,2 to one a little brighter 
than the third. Note the comma ; 3,2 (3 comma 2) must not be confounded 
with 3.2 (3 point 2): the first means 2f magnitude, the other 3/o mag- 

556. Stars Visible to the Naked Eye. — Heis enumerates the 
stars clearly visible to the naked eye in the part of the sky 
north of 35° south declination, as follows : 

1st magnitude ... 14 4th magnitude . . . 313 Number of 

2d " ... 48 5th " ... 854 visible stars 

3d " ... 152 6th " ... 2010 
Total 3391 

of the differ- 
ent m? 

ent magni- 

According to Newcomb, the number of stars of each magni- 
tude is such that united they would give, roughly speaking, 
about the same amount of light as that received from the aggre- 
gate of those of the next brighter magnitude. But the relation 
is very far from exact and fails entirely for the magnitudes 
below the eleventh, the smaller stars being much less numerous 
than this law would make them. 

557. Light-Ratio and Scale of Magnitude. — It was found by 
Sir John Herschel about 1830 that an average star of the first 
magnitude is just about one hundred thnes as bright as one of First-magni- 
the sixth, and that for the naked-eye stars a correspondinsf ratio *"^^® ^^^^ 

. . equals one 

had been roughly maintained by former observers through the hundred of 
whole scale of magnitudes, the stars of each magnitude being sixth magni- 
approximately two and one-half times as bright as those of the 
next fainter. 

Still, on the star-maps of Argelander, Heis, and others 
long accepted as standards there are notable deviations from 
a consistent uniformity, and in 1850 Pogson proposed the 

1 The term "Uranometria'' has come to mean a catalogue of naked-eye stars, 
like the catalogues of Hipparchus, Ptolemy, and Ulugh Beigh. 



The "Abso- 
lute Scale." 

a standard, 
tude star. 

of stars and 
their differ- 
ence of 

"Absolute Scale" of star magnitudes, adopting the fifth root of one 
hundred, 2.512 +, as the uniform "light-ratio," adjusting the first 
six magnitudes to fit as nearly as possible the magnitudes of 
Argelander's Durchmusterung, and then carrying forward the 
scale among the telescopic stars. Until about 1885 this scale 
had not been much used ; but it has been adopted in the new 
" Uranometrias " made at Cambridge, U.S., and Oxford, and 
is now rapidly supplanting all the arbitrary scales of former 
observers. On this scale Aldebaran and Altair are very nearly 
typical "first-magnitude" stars, and the two "pointers" and 
Polaris are practically "second-magnitude" stars. 

558. Relative Brightness of Different Star Magnitudes. — In 
this scale the "light-ratio" (i.e., the ratio between the light of 
two stars standing just one magnitude apart in the scale) is 

exactly vlOO, or the number whose logarithm is O.IfOOO, i.e., 
2.512. Its reciprocal is the number whose logarithm is 9.6000, 
viz., 0.3981. 

If J„^ is the brightness of a star of the mth magnitude 
(expressed either in candle-power or some other convenient 
unit), and if 5,^ is that of a star of the ^th magnitude, the rela- 
tion between their light is given by the fundament